ON THE EQUIVALENCE OF MEASURES ON LOOP SPACE

VIKRAM K. SRIMURTHY

Abstract. Let K be a simply-connected compact Lie Group equipped with an Ad_{K}-invariant inner product on the Lie Algebra 𝔎, of K. Given this data, there is a well known left invariant ”H^{1}-Riemannian structure” on L (K) (the infinite dimensional group of continuous based loops in K), as well as a heat kernel ν_{T} (k_{0}, ·) associated with the Laplace-Beltrami operator on L (K). Here T > 0, k_{0} ∈ L (K), and ν_{T} (k_{0}, ·) is a certain probability measure on L (K). In this paper we show that ν_{1} (e, ·) is equivalent to Pinned Wiener Measure on K on 𝔊_{s_{0}} ≡ σ ⟨x_{t} : t ∈ [0, s_{0}]⟩ (the σ-algebra generated by truncated loops up to “time” s_{0})

Contents

1. Introduction 2 2. Statement of Results 3 2.1. Loop group Geometry 3 2.2. Measures on the Loop group 5 2.3. The stochastic framework 10 3. Warm-up Section: 13 3.1. Path group cases for a Lie group: 13 3.2. Semimartingale Properties of X_{ts}: 19 3.3. Abelian Loop group Examples 25 4. The Airault-Malliavin Theorem 32 5. Absolute continuity of Heat Kernel with respect to pinned Wiener

measure 40 6. Semi-Martingale Properties of g_{T,·} 48 6.1. Preliminaries 48 6.2. The Main Theorem 52 6.3. Proof of Theorems 6.16 and 6.17 56 6.4. Propositions 6.13, 6.19, 6.20, 6.21 60 6.5. Good partitions(proof of Lemma 6.15) 71 7. HKM ↓_{𝔉_{s}}∽ P W M ↓_{𝔉_{s}} 77 8. Appendix 82 8.1. General Technical results 82 8.2. Brownian Sheets and bridges 84 8.3. Proof of Lemma 8.8 88 8.4. Gaussian Measures 90 9. Conjectures on Gaussian measure equivalence 97 References 100

This paper is in final form and no version of it will be submitted for publication elsewhere.

1. Introduction

In this paper we consider the equivalence of two measures on the Loop space of a compact Lie group. This so-called “Loop group” is the space of continuous paths in the Lie group based at the identity equipped with a certain well-known left-invariant “H^{1}-Riemannian structure”. The study of Loop groups is motivated primarily by physics and the theory of group Representations. They have been studied extensively in both the mathematics and the physics literature. See for example [29], [19], [27],[3], [15], [16], [1], [24], [18], [12] and the references therein.

Heat Kernel and pinned Wiener measure are two natural measures that have been advocated as the “right” measure on the Loop groups. Pinned Wiener measure on a Loop group is the law of a group-valued Brownian motion that has been conditioned on loops. This measure has been extensively studied in [17], [25], [2], [26]. Heat Kernel measure has been studied in [13], [11] as another natural measure on Loop Space. In [13], Driver and Lohrenz showed that there exists a certain process that deserves to be called “Brownian motion” on the path space of a Loop group. The Heat Kernel measures on the Loop Space are the time t, t > 0 distributions of this Brownian motion. Thus it is a natural question to consider the equivalence of these two measures.

A further motivation comes from logarithmic Sobolev inequalities and the papers of Getzler [16], Gross [17], Driver [11], Hsu, Aida, and Elworthy. The classical Sobolev inequalities are a fundamental tool in analyzing finite-dimensional manifolds. For infinite-dimensional manifolds logarithmic Sobolev inequalities, because of their dimension-independent character, are seen to be the proper analogues of classical Sobolev inequalities. Logarithmic Sobolev inequalities have been studied extensively over infinite-dimensional linear spaces as well as finite-dimensional manifolds (see [8], [9] for surveys and [20]). If a logarithmic Sobolev inequality does hold for pinned Wiener measure, μ_{0}, then the Dirichlet form μ_{0} ⟨∇f, ∇f⟩ associated with pinned Wiener measure will have a spectral gap (the so-called “Mass Gap inequality”).

In [16], Getzler showed that the Bakry and Emery criteria (see [4] and [5]) for proving a logarithmic Sobolev inequality does not hold in general for loop groups when the “underlying measure” is pinned Wiener measure. In [17], using pinned Wiener measure, Gross showed that a logarithmic Sobolev inequality on Loop space does hold, but with an added potential term (a so-called “defective” logarithmic Sobolev inequality). Using Heat Kernel measure instead, Driver and Lohrenz proved in [13] that a logarithmic Sobolev inequality does hold on Loop groups, without Gross’ potential. If Heat Kernel and pinned Wiener measures were equivalent with Radon-Nikodym derivatives bounded above and below then the Holley-Stroock Lemma (see [20])would tell us that pinned Wiener measure admits a classical (i.e. “non-defective”) logarithmic Sobolev inequality. Even if the equivalence were not so nice, it might still be possible to use the Driver-Lohrenz result of [13] to eliminate the Gross’ potential term and thereby prove a logarithmic Sobolev inequality for pinned Wiener measure.

In Section 5, using a result of Malliavin and Airault (see [26] and Theorem 4.1) as well as a maximal-inequality argument, we show that Heat Kernel measure is absolutely continuous with respect to pinned Wiener measure. Further, the relevant Radon-Nikodym derivative is bounded. We also provide a simpler and more direct proof of the result of the Malliavin-Airault Theorem in Section 4.

In Section 7 we show that pinned Wiener measure is absolutely continuous (and thus equivalent) with respect to Heat Kernel measure on 𝔉_{s} (𝔉_{s} denotes the σalgebra of functions depending on the loop up to time s < 1). We view the Loop-Space-valued Brownian motion, developed by Driver and Lohrenz in [13], as a group-valued two-parameter process. Viewing one of the parameters fixed, the resulting process has the same distribution as Heat Kernel measure. In Section 6 we show that, in the other parameter, this process is a Brownian semimartingale on the path space of the Lie group. To do this, we use extensively the theory of two-parameter semimartingales developed by Cairoli, Walsh, Wang, and Zakai (see [7], [31]). The fact that we can pull back this process to a Lie algebra valued Brownian Semimartingale, Girsanov’s Theorem, and the fact that Wiener measure and pinned Wiener measure are equivalent on 𝔉_{s}; gives us our result that on 𝔉_{s} Heat Kernel measure and pinned Wiener measure are equivalent. In our proof, the analysis is done in a bigger space (the Wiener space of the compact Lie group) which is why we require s to be strictly less than one.

Heat Kernel measure is a time t distribution of a process on the path space of a Loop group which is started from the identity loop (i.e. the constant loop). This describes a homotopy between the endpoint of this process and the identity loop. As a consequence, Heat Kernel measure concentrates all its mass on nullhomotopic loops. On the other hand pinned Wiener measure is quasi-invariant under translations by finite-energy loops. Thus Pinned Wiener measure must assign non-zero mass to all homotopy classes. Therefore if the Lie group is not simply connected, pinned Wiener measure is not equivalent to Heat Kernel measure. Thus our result showing absolute continuity on 𝔉_{s} for s < 1 is in a sense the best result that can be obtained in the non-simply-connected case.

In our last section, Section 9, we conjecture that pinned Wiener measure is absolutely continuous with respect to a weighted sum of Heat Kernel measures on the various homotopy classes. These Heat Kernel measures are obtained by starting the Driver-Lohrenz Loop-group-valued Brownian motion at the energy-minimizing geodesics in each homotopy class. This results in a measure that assigns non-zero mass to each homotopy class. The conjecture rests on a very informal computation done by Driver and the fact that the conjecture is true in the case that the compact Lie group is the circle S^{1}.

2. Statement of Results

2.1. Loop group Geometry. Let K be a connected compact Lie group, 𝔎 ≡ T_{e}K be the Lie algebra of K, and ⟨·, ·⟩_{𝔎} be an Ad_{K}-invariant inner product on 𝔎. For ξ ∈ 𝔎, let |ξ|_{𝔎} ≡ \sqrt{⟨ξ, ξ⟩_{𝔎}}. Let ℓ_{g} and ρ_{g} be left and right translations on K respectively. (i.e. ℓ_{g} and ρ_{g} are maps taking K to K so that ℓ_{g} (x) = gx while ρ_{g} (x) = xg). Let

L (K) ≡ {σ ∈ C ([0, 1] → K) |σ (0) = σ (1) = e}

denote the based loop group on K consisting of continuous paths σ : [0, 1] → K such that σ (0) = σ (1) = e, where e ∈ K, is the identity element.

Definition 2.1 (Tangent Space of L (K)). We will need the following definitions:-

• Given a function h : [0, 1] → 𝔎 such that h (0) = 0, define (h, h)_{H} = ∞ if h is not absolutely continuous and set (h, h)_{H} = ∫_{0}^{1} |h′ (s)|^{2} ds otherwise.

• Define

H (𝔎) ≡ {h : [0, 1] → 𝔎|h (0) = 0 and (h, h) < ∞} .

Then H (𝔎) is a Hilbert space under (·, ·)_{H}.

• Define

H_{0} (𝔎) ≡ {h ∈ H (𝔎) |h (1) = 0} .

Then (H_{0} (𝔎) , (·, ·)_{H}) is also a Hilbert space.

In order to define the tangent space T L (K) of L (K) let θ denote the Maurer-Cartan form. That is θ ⟨ξ⟩ ≡ (ℓ_{k^{−1}})_{∗} ξ for all ξ ∈ T_{k}K, and k ∈ K and where ℓ_{g} denotes left multiplication by g ∈ K. Let θ ⟨X⟩ (s) ≡ θ ⟨X (s)⟩ and p : T K → K be the canonical projection. We now define

T L (K) ≡ {X : [0, 1] → T K|θ ⟨X⟩ ∈ H_{0} and p ∘ X ∈ L (K)} .

By abuse of notation, use the same p to denote the canonical projection from T L (K) → L (K). As usual, define the tangent space at k ∈ L (K) by T_{k}L (K) ≡ p^{−1} {k}. Using left translations, we extend the inner product (·, ·)_{H_{0}} on H_{0} to a Riemannian metric on T L (K). Explicitly set

(X, X)_{L(K)} ≡ (θ ⟨X⟩ , θ ⟨X⟩)_{H_{0}(𝔎)} where X ∈ T L (K) .

In this way, L (K) is to be thought of as an infinite-dimensional Riemannian manifold. Viewing the Lie algebra (𝔎, 0) as a Lie group in its own right with Lie algebra 𝔎, we obtain definitions for

L (𝔎) ≡ {σ ∈ C ([0, 1] → 𝔎) |σ (0) = σ (1) = 0}

as the “Lie group” with Lie algebra H_{0} (𝔎) thought of as a commutative Lie algebra.

Definition 2.2 (Good Orthonormal basis of H_{0}). An orthonormal basis {η_{k}}_{k∈ℕ} of H_{0} (𝔎) is a good orthonormal basis if the Lie Bracket [η_{k} (s) , η′_{k} (s)] is identically zero for all values of s and k.

Example 2.3. We will provide a couple of examples for illustration:-

1. Take {h_{k}} to be an orthonormal basis of H_{0} (ℝ) and let {A} run through an orthonormal basis of 𝔎. Then η_{A,k} ≡ h_{k}A is a good orthonormal basis.

2. Let {η_{A,k}}_{k∈ℕ,A} be loops in H_{0}^{1} (𝔎) where

η_{A,2k} (τ) ≡ \frac{A}{πk\sqrt{2}} sin 2πkτ
η_{A,2k−1} (τ) ≡ \frac{A}{πk\sqrt{2}} (cos 2πkτ − 1)

and A runs through an orthonormal basis of 𝔎.

Definition 2.4 (The Laplacian △_{L(K)} and △_{L(𝔎)}). Take a good orthonormal basis of H_{0} (𝔎). Then define an operator △_{L(K)} on functions f on L (K) by setting

△_{L(K)}f ≡ ∑ ∂_{h}^{2}f,

where

(∂_{h}f) (γ) ≡ ∂_{ε}f (γ exp εh) |_{ε=0}.

Define the Laplacian △_{L(𝔎)} on functions f on L (𝔎) in the same way above by setting

△_{L(𝔎)}f ≡ ∑ ∂_{h}^{2}f,

where

(∂_{h}f) (γ) ≡ ∂_{ε}f (γ + εh) |_{ε=0}.

Definition 2.5 (Cylinder functions). Let (R, e) denote either the Lie group (K, e) or the Lie algebra (𝔎, 0). Let L̃ (R) denote either L (R) or W_{e} (R).

1. Then f : L̃ (R) → ℝ is a cylinder function iff f (σ) ≡ F (σ_{t_{1}}, ⋯, σ_{t_{n}}) where {0 < t_{1} < ⋯ < t_{n} < 1}. F ∈ C (R^{n}).

2. f is a smooth cylinder function iff F ∈ C^{∞} (R^{n}). F ∈ C (R^{n}).

3. f is a bounded cylinder function iff F ∈ C_{b} (R^{n}). Here C_{b} (R^{n}) are the bounded continuous functions on R^{n}.

4. Let FC (L̃ (R)) denote the space of all cylindrical functions.

5. Let FC^{∞} (L̃ (R)) denote the space of all smooth cylindrical functions.

6. Let FC_{b}^{∞} (L̃ (R)) denote the space of all bounded cylindrical functions.

7. A cylinder function is 𝔉_{ℙ}-measurable if and only if f (σ) = F (σ_{t_{1}}, ⋯, σ_{t_{n}}) where {t_{i}} ⊂ ℙ where ℙ is some partition of [0, 1] .

2.2. Measures on the Loop group.

2.2.1. Pinned Wiener measure. Let the Wiener space W_{e} (K) denote the space of all continuous paths in K starting at the identity. Explicitly

W_{e} (K) ≡ {σ ∈ C ([0, 1] → K) |σ (0) = e} .

Definition 2.6 (Heat Kernel measure on K). Let t > 0. The Heat Kernels P_{t}^{K} on K are the unique functions so that for any smooth f on K, the function u on [0, ∞) × K defined by setting u (t, x) ≡ ∫_{K} f (y) P_{t}^{K} (y^{−1}x) dy is a solution to the Heat equation with initial condition f. Explicitly

∂_{t}u = \frac{1}{2}△_{K}u
u (t, x) → f (x) as t → 0.

It is well known that x → P_{t}^{K} are smooth function on K and that P_{t}^{K} (x) = P_{t}^{K} (x^{−1}).

Definition 2.7 (Wiener Measure on W_{e} (K)). Wiener Measure, μ_{t}, on W_{e} (K) with parameter t, is the unique measure so that for any bounded cylinder function f of the form f (x) = F (x_{s_{1}}, ⋯, x_{s_{n}}) we have

μ_{t} [f] ≡ ∫_{K^{n}} F (x_{1}, ⋯, x_{n}) ∏_{i=1}^{n} P_{t(s_{i}−s_{i−1})}^{K} (x_{i−1}^{−1}x_{i}) dx_{i},

where x_{0} = e and s_{0} = 0. [The measure μ_{1} will also be denoted by μ in the sequel.]

Definition 2.8 (Brownian motion on K). We will state three equivalent definitions. A process s → β (s) is a Brownian motion on K starting at e with parameter t iff:- 1. β is a W_{e} (K)-valued random variable distributed according to Wiener mea-

sure μ_{t}.

2. the process s → β (s) is a diffusion starting at e with generator \frac{t}{2}△_{K}. This means that the process s → β (s) is a martingale so that β (0) = e a.s. and

(ϕ ∘ β) (ds) = ϕ′ ∘ β (s) β (ds) + \frac{t}{2} (△_{K}ϕ) ∘ β (s) ds

for any smooth ϕ on K. Here △_{K} is the Laplacian on K with respect to the metric ⟨·, ·⟩_{K} on K.

The first definition is easier in simpler cases like ℝ^{d} or compact Lie groups. The second definition is easier to extend to the infinite-dimensional cases and manifolds. See Definitions 2.14 and 3.6.

Definition 2.9 (Pinned Wiener Measure). Pinned Wiener Measure, μ_{0,t}, on L (K) with parameter t is the unique measure on L (K) so that for any bounded cylinder functions f of the form f (x) = F (x_{s_{1}}, ⋯, x_{s_{n}}) where F ∈ C^{∞} (K), then

μ_{0,t} [f] ≡ ∫_{K^{n}} F (x_{1}, ⋯, x_{n}) \frac{P_{t(1−s_{n})}^{K} (x_{n}, e)}{P_{t}^{K} (e, e)} ∏_{i=1}^{n} P_{t(s_{i}−s_{i−1})}^{K} (x_{i−1}, x_{i}) dx_{i}, (2.1)

where x_{0} = e and s_{0} = 0.[We will use the notation μ_{0} to denote μ_{0,1}.]

Remark 2.10 (Pinned Wiener measure is really pinned!). Pinned Wiener measure is really Wiener measure pinned at e. At least on cylinder functions,

μ_{0,t} [f] = ∫ f (x) δ_{e} (x (1)) μ_{t} (dx) / ∫ δ_{e} (x (1)) μ_{t} (dx) .

As Malliavin showed, another way of looking at this measure is

μ_{0} (f) ≡ (\frac{d (π_{1})_{∗} (fμ_{t})}{d (π_{1})_{∗} μ_{t}}) (e) , (2.2)

where π_{1} : x_{·} → x_{1}; μ_{t} is Wiener measure on K with parameter t; and (fμ_{t}) is that measure on W_{e} (K) so that (fμ_{t}) (dx) = f (x) μ_{t} (dx). For cylinder functions, it is trivial to check Eq. [2.2] by writing down finite-dimensional distributions.

Definition 2.11 (Brownian bridge on K). s → χ (s) is a Brownian bridge from on K with parameter t if χ is an L (K)-valued random variable distributed according to pinned Wiener measure μ_{0,t}.

2.2.2. Heat Kernel measure.

Definition 2.12 (Brownian Bridge Sheet on 𝔎). A Gaussian process {χ (t)}_{t∈[0,1]} is a Brownian bridge Sheet on 𝔎 if for (t, s) in [0, 1]^{2}, χ (t, s) is a 𝔎-valued mean-zero Gaussian process with covariance given by

E ⟨A, χ (t, s)⟩_{𝔎} ⟨B, χ (τ, σ)⟩_{𝔎} = ⟨A, B⟩_{𝔎} (t ∧ τ) G_{0} (s, σ) ,

where χ (t, s) ≡ χ (t) (s) ∈ 𝔎; A, B ∈ 𝔎; t, τ, s, σ ∈ [0, 1]; and G_{0} (s, σ) ≡ s ∧ σ − sσ.

Remark 2.13. It turns out that if β is a Brownian bridge sheet on 𝔎 then β_{ts} is continuous in both its parameters, t → β_{ts} is a Brownian motion on 𝔎 with parameter G_{0} (s, s) and s → β_{ts} is a Brownian bridge on 𝔎 with parameter s.

Definition 2.14 (Brownian motion on L (K)). A process t → Σ (t, ·) is an L (K)- valued Brownian motion if and only if for any smooth cylinder function f : L (K) → ℝ, there is a real-valued martingale M_{t} so that

f (Σ (dt, ·)) = dM_{t} + \frac{1}{2} (△_{L(K)}f) (Σ (t, ·)) dt.

See Theorem 2.19 for the existence of this Brownian motion. So t → Σ (t, ·) is a diffusion on L (K) with generator \frac{1}{2}△_{L(K)}. [Define a Brownian motion on L (𝔎) by thinking of 𝔎 as a Lie group and applying the above definition]

Lemma 2.15 (△_{L(K)} on cylinder functions, see [13]). Let G_{0} be as in Definition 2.12. Let ℙ be the partition {0 < s_{1} < ⋯ < s_{n} < 1}. Let π_{ℙ} be the map taking a loop σ in L (K) to (σ_{s_{1}}, ⋯, σ_{s_{n}}) ∈ K^{n}. For F ∈ C^{∞} (K^{n}), define

(A^{(i)}F ) (g_{1}, ⋯, g_{n}) ≡ \frac{d}{dt}|_{t=0}F (⋯, g_{i} exp tA, ⋯) .

Define an elliptic operator △_{ℙ} on C^{∞} (K^{n}) by setting

△_{ℙ} ≡ ∑_{i,j,A} G_{0} (s_{i}, s_{j}) A^{(i)}A^{(j)}.

Then letting A run through an orthonormal basis of 𝔎, for any smooth cylinder function F : K^{n} → ℝ we have

△_{L(K)} (F ∘ π_{ℙ}) = (△_{ℙ}F) ∘ π_{ℙ}.

[This Lemma can also be used on the Lie algebra 𝔎 by viewing 𝔎 itself as a Lie group, i.e. take K = 𝔎 and g exp A = g + A. ]

Proof. As in Example 2.3, take {h_{k}} to be an orthonormal basis of H_{0} (ℝ) and let {A} run through an orthonormal basis of 𝔎. Then η_{A,k} ≡ h_{k}A is a good orthonormal basis of H_{0} (𝔎). Then we have

(∂_{η_{A,k}}F ∘ π_{ℙ}) (γ) = \frac{d}{dt}F ∘ π_{ℙ} (γ exp tη_{A,k}) ↓_{0}
= ∑_{i} η_{k} (s_{i}) (A^{(i)}F ) ∘ π_{ℙ} (γ) .

Thus

△_{L(K)}F ∘ π_{ℙ} = ∑_{k∈ℕ,A} ∂_{η_{A,k}}∂_{η_{A,k}}F ∘ π_{ℙ}

= ∑_{i=1}^{n} ∑_{k∈ℕ,A} η_{k} (s_{i}) ∂_{η_{A,k}} (A^{(i)}F ) ∘ π_{ℙ} (γ)

= ∑_{A} ∑_{i,j=1}^{n} [∑_{k∈ℕ} η_{k} (s_{i}) η_{k} (s_{j})] (A^{(j)}A^{(i)}F ) ∘ π_{ℙ} (γ) .

It remains only to show that ∑_{k∈ℕ} η_{k} (s_{i}) η_{k} (s_{j}) = G_{0} (s_{i}, s_{j}). Let h ∈ H_{0} (ℝ). Suppose we can show that

⟨G_{0} (s, ·) , h⟩_{H_{0}(ℝ)} = h (s) . (2.3)

A priori we suspect that such elements G_{0} (s, ·) exist because the evaluation map h → h_{s} is a bounded linear functional on the Hilbert Space H_{0} (ℝ). Then we will be done since we shall have

∑_{k∈ℕ} η_{k} (s_{i}) η_{k} (s_{j}) = ∑_{k∈ℕ} ⟨G_{0} (s_{i}, ·) , η_{k}⟩ ⟨G_{0} (s_{j}, ·) , η_{k}⟩
= ⟨G_{0} (s_{i}, ·) , G_{0} (s_{j}, ·)⟩
= G_{0} (s_{i}, s_{j}) .

We shall proceed to check Eq. [2.3].

⟨G_{0} (s, ·) , h⟩_{H_{0}(ℝ)} = ∫_{0}^{1} ∂_{t} (s ∧ t − st) h′ (t) dt
= ∫_{0}^{1} (1_{[0,s]} (t) − s) h′ (t) dt
= h (s) − (1 − s) h (0) − sh (1)
= 0 since h is a loop based at 0.

Hence we are done.

Lemma 2.16 (Brownian Motion on L (𝔎) exists). If χ_{t} ≡ χ (t, ·) is a Brownian Sheet then for any smooth cylindrical function f, there is a real-valued martingale M_{t} so that

df (χ_{t}) = dM_{t} + \frac{1}{2} (△_{L(𝔎)}f) (χ_{t}) dt.

Here △_{L(𝔎)} is the Laplace-Beltrami operator defined in Definition 2.4. So every Brownian bridge Sheet on 𝔎 is an L (𝔎)-valued Brownian motion.

Proof. Let G_{0} be as in Definition 2.12. Let χ_{ts}^{A} denote as usual ⟨χ_{ts}, A⟩_{𝔎} for any A ∈ 𝔎. Then the joint quadratic variation χ_{dts}^{A}χ_{dtσ}^{B} = ⟨A, B⟩_{𝔎} G_{0} (s, σ) dt. Let f be a smooth cylinder function implies (see Definition 2.5), f (σ) = F (σ_{s_{1}}, ⋯, σ_{s_{n}}) where F ∈ C^{∞} (𝔎^{n}) and ℙ ≡ {0 < s_{1} < ⋯ < s_{n} < 1}. Let χ_{t}^{ℙ} denote (χ_{ts_{1}}, ⋯, χ_{ts_{n}}). Let (A^{(i)}F ) (g_{1}, ⋯, g_{n}) denote ∂_{ε}F (⋯, g_{i} + εA, ⋯) ↓_{ε=0}. Let △_{ℙ} on 𝔎^{n} by

△_{ℙ} ≡ ∑_{i,j,A} G_{0} (s_{i}, s_{j}) A^{(i)}A^{(j)}.

Thus by Ito’s Lemma we have

df (χ_{t}) = dF (χ_{t}^{ℙ})
= ∑_{i,A} (A^{(i)}F ) (χ_{t}^{ℙ}) χ_{dts_{i}}^{A}

+\frac{1}{2} ∑_{i,j,A} (A^{(j)}A^{(i)}F ) (χ_{t}^{ℙ}) χ_{dts_{i}}^{A}χ_{dts_{j}}^{A}

= dMartingale + \frac{1}{2} ∑_{i,j,A} G_{0} (s_{i}, s_{j}) (A^{(j)}A^{(i)}F ) (χ_{t}^{ℙ}) dt

= dMartingale + \frac{1}{2} (△_{ℙ}F) (χ_{t}^{ℙ}) dt.

By Lemma 2.15 (view 𝔎 as a Lie algebra in its own right with Lie algebra 𝔎 while applying this Lemma) this last expression is just

= dMartingale + \frac{1}{2} (△_{L(K)}f) (χ_{t}) dt.

We will need the the following Theorem:

Theorem 2.17 (Malliavin). Let (Ω_{0}, 𝔉^{0}, {𝔉_{ts}^{0}}_{(t,s)∈[0,1]^{2}} , P_{0}) be a filtered complete probability space where

𝔉_{ts}^{0} ≡ σ ⟨χ_{τu} : τ ∈ [0, t] , u ∈ [0, s]⟩ ,

and 𝔉^{0} ≡ ∨_{(t,s)∈[0,1]^{2}}𝔉_{ts}^{0}. Let k_{0} ∈ L (K) and let χ be a 𝔎-valued Brownian bridge sheet in the sense of Definition 2.12. Recall ℓ_{g} : K → K takes x → gx. Then there is a jointly continuous solution Σ (t, s) to the stochastic differential equation

Σ (δt, s) = ∑_{A∈ONB(𝔎)} (ℓ_{Σ(t,s)∗}A) χ^{A} (δt, s) with Σ (0, s) = k_{0} (s) , ∀s ∈ [0, 1] , (2.4)

where the A run through an orthonormal basis of 𝔎 and where for each fixed s ∈ [0, 1] , Σ (δt, s) and χ^{A} (δt, s) denote the Fisk-Stratonowicz differentials of the processes t → Σ (t, s) and t → ⟨χ (t, s) , A⟩_{𝔎} respectively. Henceforth we write Eq. (2.4) more concisely as

Σ (δt, s) = (L_{Σ(t,s)})_{∗} χ (δt, s) with Σ (0, s) = k_{0} (s) , ∀s ∈ [0, 1] . (2.5)

[see Malliavin [27]; see also Theorem 3.8 of [11]]

Remark 2.18 (Explicit Matrix Representation of Eq. [2.5). ] Let M_{m} (ℝ) be all m× m matrices on ℝ and GL_{m} (ℝ) be all invertible matrices in M_{m} (ℝ). We will work with an explicit matrix representation of our Lie group K. K will be thought of as a subgroup of GL_{m} (ℝ) ⊂ M_{m} (ℝ) for some m. Such a representation exists as a consequence of the Peter-Weyl Theorem. Hence Eq. (2.5) can be rewritten as

Σ (δt, s) = Σ (t, s) χ (δt, s) with Σ (0, ·) = k_{0}, ∀s ∈ [0, 1] , (2.6)

where we have used matrix multiplication to define Σ (t, s) χ (δt, s). Explicitly if we let B_{ij} denote the i, j entry of the matrix B we have

δ_{t} (Σ (t, s))_{ij} = ∑_{k} (Σ (t, s))_{ik} δ_{t} (χ (t, s))_{kj} .

Theorem 2.19 (Brownian motion on L (K)). Let Σ (t, s) be the process from Theorem 2.17 and Remark 2.18. Theorem 2.17 tells us that s → Σ (t, s) is a Loop a.s. Let Σ_{t} denote this loop s → Σ (t, s). Then t → Σ_{t} is a Brownian motion on L (K) in the sense of Definition 2.14.

Proof. See Theorem 3.10 of Driver [11].

Now that we know that Brownian motion on L (K) exists, we can define Heat Kernel measure on L (K).

Definition 2.20 (Heat Kernel measure on L (K)). Let k_{0} ∈ L (K) be a loop and let t > 0. Let Σ (t, ·) be an L (K)-valued Brownian motion so that Σ (0, ·) = k_{0} in L (K) a.s. Then, as in the finite-dimensional manifold case, Heat Kernel measure ν_{t} (k_{0}, dk) is defined to be the law of Σ (t, ·). Explicitly

∫_{L(K)} f (k) ν_{t} (k_{0}, dk) = Ef ∘ Σ (t, ·) .

The next Theorem shows that Heat Kernel measures behave as expected, in that they may be used to solve the Heat Equation on L (K). See Remark 3.12 for motivation of Theorem 2.21.

Theorem 2.21 (Driver&Lohrenz). For each t > 0, for all bounded cylinder functions f on L (K), the function u on (0, ∞) × L (K) given by

u (t, k_{0}) ≡ ∫_{L(K)} f (k) ν_{t} (k_{0}, dk) ,

is the unique solution to the heat equation

∂u (t, ·) /∂t = \frac{1}{2}△_{L(K)}u (t, ·) with lim_{t↓0} u (t, k) = f (k_{0}) .

Here △_{L(K)} denotes the operator from Definition 2.4. See Theorem 1.1 of [13]. See also Definitions 3.10 and 4.17 in [13]. [Note:- In [13], results on Heat kernel measures are obtained for groups of compact type, and not merely compact Lie groups.]

2.3. The stochastic framework. We shall use the results of Section 2.2.2 to obtain our probability space.

Definition 2.22 (Ambient probability space). (Ω, 𝔉, {𝔉_{ts}}_{(t,s)∈[0,1]^{2}} , P ) is going to be our biparametrically-filtered probability space where

• Ω ≡ C ([0, 1] → L (K)) equipped with 𝔉, the completion of the Borel σalgebra.

• Let Σ be the process from Theorem 2.17 so that Σ_{0} = e, where e denotes the identity loop.

• P is defined to be Wiener Measure on C ([0, 1] → L (K)). Explicitly, P ≡ Law Σ.

• g_{t} : C ([0, 1] → L (K)) → L (K) by x → x (t) for any x ∈ C ([0, 1] → L (K))

• By Theorem 2.19 we see that dLaw g_{t} = dν_{t} (e, ·).

• g_{ts} (x) = [x (t)] (s) in K.

• 𝔉_{00} is a σ-algebra containing all the null sets of 𝔉.

• 𝔉_{ts} ≡ σ ⟨g_{τσ} : τ ∈ [0, t] and σ ∈ [0, s]⟩ ∨ 𝔉_{00}.

Definition 2.23. Let {𝔊_{s}} be a filtration. Then U_{·} is a K-valued 𝔊_{·}-semimartingale iff for any smooth f : K → ℝ the process t → f (U_{t}) is an ℝ-valued 𝔊_{·}-semimartingale.

Definition 2.24 (see Protter [30]). Let {𝔊_{s}} be a filtration. An ℝ-valued process U_{·} is called an ℝ-valued 𝔊_{·}-semimartingale if:-

1. the paths U_{·} are continuous a.s.

2. U_{·} is adapted with respect to the filtration 𝔊_{·} (i.e. U_{t} ∈ 𝔊_{t} for all t ∈ [0, T])

3. Given any sequence of simple adapted processes {H} then ∫_{0}^{T} H_{t}U_{dt} ↓ 0 in probability whenever H ↓ 0 uniformly on compacts in probability. Here H is a simple adapted process if H is an ℝ-valued 𝔊_{t}-adapted process of the form H (t, ω) ≡ ∑_{i=0}^{n} H_{i} (ω) 1_{(T_{i},T_{i+1}]} with the T_{i} being a sequence of stopping times with 0 ≤ T_{0} ≤ ⋯ ≤ T_{n} ≤ T. The integral ∫_{0}^{T} H_{t}U_{dt} is defined to be

the sum ∑_{i=0}^{n} H_{i} (U_{T_{i+1}} − U_{T_{i}}) for any simple adapted process H.

Theorem 2.25 (Semimartingale properties of g_{·s}). The process g of Definition 2.22 has the following properties:-

1. The process t → g_{ts} is a semimartingale.

2. Let X_{ts} ≡ ∫_{0}^{t} g_{τs}^{−1}g_{δτs}. Then t → X_{t·} is a Brownian bridge sheet on 𝔎 with respect to the measure P. Furthermore, X can be taken to be continuous in both its parameters.

Remark 2.26. After the proof of Theorem 2.25 we shall never again refer to χ, Σ or the underlying abstract probability space. Also we will always use the version of X that is continuous in both parameters t and s.

Proof. of Theorem 2.25

First we check that t → g_{ts} is an 𝔉_{·s}-semimartingale. For convenience we use the “good integrator” definition of a semimartingale (see Definition 2.23). Pick f ∈ C^{∞} (K). It will suffice to check that f (g_{·s}) is a semimartingale. Let {H} be a sequence of 𝔉_{·s}-adapted processes which converge to zero uniformly on compacts in probability. Then we have

P (|∫_{0}^{T} H_{t}g_{dts}| > ε) = P (|∑_{i=0}^{n} H_{i} (g_{T_{i+1}s} − g_{T_{i}s})| > ε)

= P ({ω : |∑_{i=0}^{n} H_{i} (Σ (ω)) (Σ_{T_{i+1}s} − Σ_{T_{i}s})| > ε})

= P ({ω : |∫_{0}^{T} H_{t} ∘ Σ_{·s} (ω) Σ_{dts}| > ε}) .

This last term goes to zero since t → Σ_{ts} is an 𝔉_{·s}-semimartingale. Thus g_{·s} is a semimartingale.

Now we want to show that X_{·s} ≡ ∫_{0}^{·} g_{ts}^{−1}g_{δts} has the same law as χ_{·s}. Let E_{ij} denote the m × m matrix with k, l-entries δ_{ik}δ_{jl}. We can write

X_{·s} = ∑_{i,j,k} ∫_{0}^{·} (g_{ts}^{−1})_{ik} δ_{t} (g_{ts})_{kj} E_{ij}

= ∑_{j,k} ∫_{0}^{·} [∑_{i} (g_{ts}^{−1})_{ik} E_{ij}] δ_{t} (g_{ts})_{kj} ,

and

χ_{·s} = ∑_{i,j,k} ∫_{0}^{·} (Σ_{ts}^{−1})_{ik} δ_{t} (Σ_{ts})_{kj} E_{ij}

= ∑_{j,k} ∫_{0}^{·} [∑_{i} (Σ_{ts}^{−1})_{ik} E_{ij}] δ_{t} (Σ_{ts})_{kj} .

Thus we can write

X_{·s} = ∑_{k} ∫_{0}^{·} f_{k} (g_{ts}) δ_{t}h_{k} (g_{ts}) ,

and

χ_{·s} = ∑_{k} ∫_{0}^{·} f_{k} (Σ_{ts}) δ_{t}h_{k} (Σ_{ts}) ;

where f_{k} and h_{k} are matrix-valued and ℝ-valued functions on K respectively. In particular, by the definition of the Fisk-Stratonowicz integral, X_{T s} is the limit (in probability with respect to measure P) of the sequence

X_{T s}^{ℙ} ≡ ∑_{k,ℙ} ∫_{0}^{·} \frac{1}{2} [f_{k} (g_{t_{i−1}s}) + f_{k} (g_{t_{i}s})] [h_{k} (g_{t_{i}s}) − h_{k} (g_{t_{i−1}s})] ,

and χ_{T s} is the limit (in probability with respect to the measure P) of the sequence

χ_{T s}^{ℙ} ≡ ∑_{k,ℙ} ∫_{0}^{·} \frac{1}{2} [f_{k} (Σ_{t_{i−1}s}) + f_{k} (Σ_{t_{i}s})] [h_{k} (Σ_{t_{i}s}) − h_{k} (Σ_{t_{i−1}s})] .

Now

P ({ω : |χ_{T s} − X_{T s} ∘ Σ| > ε}) = P ({ω : |χ_{T s}^{ℙ} − X_{T s} ∘ Σ| > ε})
= P ({ω : |X_{T s}^{ℙ} ∘ Σ − X_{T s} ∘ Σ| > ε})
= P (|X_{T s}^{ℙ} − X_{T s}| > ε) → 0.

Thus χ_{ts} = X_{ts} ∘ Σ almost surely ω. By continuity of χ and X in both their parameters, we have χ = X ∘ Σand therefore t → X_{t·} is a Brownian bridge sheet on 𝔎 with respect to P.

We have only to assert that a biparametrically continuous version of X can be chosen. By Theorem 8.2 it suffices to check that

P [|X_{ts} − X_{τσ}|_{𝔎}^{ε}] ≤ C [(t − τ)^{2} + (s − σ)^{2}]^{\frac{m+β}{2}} ,

for some positive ε, C, β and m = dim 𝔎. Let t > τ.

X_{ts} − X_{τσ} = (X_{ts} − X_{τs}) + (X_{τs} − X_{τσ}) .

As in the proof of Lemma 8.3, if a martingale M has independent increments, then its quadratic variation ∫ dM_{t}dM_{t} is given by ∫ d_{t}EM_{t}^{2}. The process t → X_{ts} −X_{τs} is a Brownian motion on 𝔎 with parameter G_{0} (s, s) and so has quadratic variation (t − τ) G_{0} (s, s). The process τ → X_{τs}^{A} − X_{τσ}^{A} is also a martingale with independent increments and so has quadratic variation

∫_{0}^{τ} (X_{dus}^{A} − X_{duσ}^{A})^{2} = τ [G_{0} (s, s) + G_{0} (σ, σ) − 2G_{0} (s, σ)] .

Thus by Burkholder’s inequality we see that

P |X_{ts} − X_{τs}|^{ε} ≤ C_{m} ∑_{A} P |X_{ts}^{A} − X_{τs}^{A}|^{ε}
≤ C_{ε,m} ∑_{A} |(t − τ) G_{0} (s, s)|^{ε/2}
≤ C_{ε,m} |(t − τ)|^{ε/2} .

where the constant C_{ε,m} depends only on ε and m. Again by Burkholder, we have the estimate

E |X_{τs} − X_{τσ}|^{ε} ≤ C_{m} ∑_{A} P |X_{τs}^{A} − X_{τσ}^{A}|^{ε}
≤ C_{ε,m}τ^{ε/2} |G_{0} (s, s) + G_{0} (σ, σ) − 2G_{0} (s, σ)|^{ε/2}
≤ C_{ε,m} [|s − σ|^{ε/2} + |s − σ|^{ε}] .
≤ C_{ε,m} |s − σ|^{ε/2} .

Thus

P [|X_{ts} − X_{τσ}|_{𝔎}^{ε}] ≤ C_{ε,m} [|s − σ|^{ε/2} + |(t − τ)|^{ε/2}] .

Picking ε > m + β, we are done.

We are now in a position to state the main results of this paper.

Theorem 2.27. Let K be a compact Lie group. Then Heat Kernel measure, ν_{1} (e, ·), is absolutely continuous with respect to pinned Wiener measure, μ_{0}. Furthermore, the Radon-Nikodym derivative dν_{1} (e, ·) /dμ_{0} is bounded.

Proof. This Theorem is proved as Theorem 5.1 in Section 5.

Theorem 2.28. Let s_{0} < 1 and let 𝔊_{s_{0}} ≡ σ ⟨π_{t} : t ∈ [0, s_{0}]⟩ where π_{t} : L (K) → K is the evaluation map at time t. Then pinned Wiener measure, μ_{0}, is absolutely continuous with respect to Heat Kernel measure, ν_{1} (e, ·), on the σ-algebra 𝔊_{s_{0}}.

Proof. This Theorem is proved as Theorem 7.1 in Section 7

3. Warm-up Section:

3.1. Path group cases for a Lie group: Let the Wiener space on K, the space of all continuous paths in K starting at the identity, be given by

W_{e} (K) ≡ {σ ∈ C ([0, 1] → K) |σ (0) = e} .

The goal of this section is to assert that Heat Kernel measure on W_{e} (K) and Wiener measure on W_{e} (K) are the same.

Definition 3.1 (Riemannian Structure on W_{e} (K)). Define H ≡ H (𝔎) to be the Sobolev space of functions with one L^{2}-derivative as in Definition 2.1. We will think of H as the Lie algebra of W_{e} (K). In order to define the tangent space T W_{e} (K) of W_{e} (K) let θ denote the Maurer-Cartan form. That is θ ⟨ξ⟩ ≡ (ℓ_{k^{−1}})_{∗} ξ for all ξ ∈ T_{k}K, and k ∈ K and where ℓ_{g} denotes left multiplication by g ∈ K. Let θ ⟨X⟩ (s) ≡ θ ⟨X (s)⟩ and p : T K → K be the canonical projection. We now define

T W_{e} (K) ≡ {X : [0, 1] → T K|θ ⟨X⟩ ∈ H and p ∘ X ∈ W_{e} (K)} .

By abuse of notation, use the same p to denote the canonical projection from T W_{e} (K) → W_{e} (K). As usual, define the tangent space at k ∈ W_{e} (K) by T_{k}W_{e} (K) ≡ p^{−1} {k}. Using left translations, we extend the inner product (·, ·)_{H} on H to a Riemannian metric on T W_{e} (K)). Explicitly set

(X, X)_{W_{e}(K)} ≡ (θ ⟨X⟩ , θ ⟨X⟩)_{H} where X ∈ T W_{e} (K) .

In this way, W_{e} (K) is to be thought of as an infinite-dimensional Riemannian manifold. Viewing the Lie algebra (𝔎, 0) as a commutative Lie group in its own right with Lie algebra 𝔎, we obtain definitions for

W_{0} (𝔎) ≡ {σ ∈ C ([0, 1] → 𝔎) |σ (0) = 0}

as the “Lie group” with Lie algebra H (𝔎) thought of as a commutative Lie algebra.

Definition 3.2 (Good Orthonormal basis of H). {η_{k}}_{k∈ℕ}, is a good orthonormal basis of H if {η_{k}}_{k∈ℕ}, is an orthonormal basis of H so that the Lie Bracket [η_{k} (s) , η′_{k} (s)] is identically zero for all values of s and k.

Example 3.3 (Good bases exist). Take {h_{k}} to be an orthonormal basis of H (ℝ) and let {A} run through an orthonormal basis of 𝔎. Then η_{A,k} ≡ h_{k}A is a good orthonormal basis of H.

Definition 3.4 (The Laplacian △_{W_{e}(K)}). Take a good orthonormal basis S of H (𝔎). Define an operator △ on functions on W_{e} (K) by taking

△f ≡ ∑_{h∈S} ∂_{h}^{2}f,

where

(∂_{h}f) (γ) ≡ ∂_{ε}f (γ exp εh) |_{ε=0}.

Define a Laplacian, denoted by △_{W_{0}(𝔎)}, on functions on W_{0} (𝔎) in the same way as above by taking

△_{W_{0}(𝔎)}f ≡ ∑_{h∈S} ∂_{h}^{2}f,

where

(∂_{h}f) (γ) ≡ ∂_{ε}f (γ + εh) |_{ε=0}.

It is well known that the operators △ and △_{W_{0}(𝔎)} defined above are independent of the choice of good orthonormal basis (see [13]).

Definition 3.5 (Brownian Sheet on 𝔎). A Gaussian process {β (t)}_{t∈[0,1]} is a 𝔎valued Brownian sheet if for (t, s) in [0, 1]^{2}, β (t, s) is a 𝔎-valued mean-zero Gaussian process with covariance given by

E ⟨A, β (t, s)⟩_{𝔎} ⟨B, β (τ, σ)⟩_{𝔎} = ⟨A, B⟩_{𝔎} (t ∧ τ) G (s, σ) ,

where β (t, s) ≡ β (t) (s) ∈ 𝔎; A, B ∈ 𝔎; t, τ, s, σ ∈ [0, 1]; and G (s, σ) ≡ min (s, σ).

Definition 3.6 (Brownian motion on W_{e} (K)). The process t → Σ (t, ·) is a W_{e} (K)- valued Brownian motion if and only if for any smooth cylinder function f : W_{e} (K) → ℝ, there is a real-valued martingale M_{t} so that

f (Σ (dt, ·)) = dM_{t} + \frac{1}{2} (△_{W_{e}(K)}f) (Σ (t, ·)) dt. (3.1)

We can define a Brownian motion on W_{0} (𝔎) by thinking of 𝔎 as a commutative Lie group, and using △_{W_{0}(𝔎)} instead of △_{W_{e}(K)} in Eq. [3.1].

Lemma 3.7 ( Effect of the Laplacian △_{W_{e}(K)} on cylinder functions, see [13]). Let G be as in Definition3.5. Let ℙ be the partition {0 < s_{1} < ⋯ < s_{n} < 1}. Define π_{ℙ} : σ → (σ_{s_{1}}, ⋯, σ_{s_{n}}) ∈ K^{n}. For F ∈ C^{∞} (K^{n}), define

(A^{(i)}F ) (g_{1}, ⋯, g_{n}) ≡ \frac{d}{dt}|_{t=0}F (⋯, g_{i} exp tA, ⋯) .

Define an elliptic operator L_{ℙ} on C^{∞} (K^{n}) by setting

L_{ℙ} ≡ ∑_{i,j,A} G (s_{i}, s_{j}) A^{(i)}A^{(j)}.

Then letting A run through an orthonormal basis of 𝔎, for any smooth cylinder function F : K^{n} → ℝ we have

△_{W_{e}(K)} (F ∘ π_{ℙ}) = (L_{ℙ}F) ∘ π_{ℙ}.

This Lemma can also be used on the Lie algebra 𝔎 by viewing 𝔎 itself as a commutative Lie group.

Proof. Use the same proof as that of Lemma 2.15 by replacing H_{0} (𝔎) , H_{0} (ℝ) , G_{0} (s, σ) by H (𝔎) , H (ℝ) , G (s, σ).

Lemma 3.8 (Brownian Motion on W_{0} (𝔎) exists). Every Brownian Sheet on 𝔎 is a W_{0} (𝔎)-valued Brownian motion. More precisely, if β_{t} ≡ β (t, ·) is a Brownian Sheet then for any smooth cylindrical function f, there is a real-valued martingale M_{t} so that

df (β_{t}) = dM_{t} + \frac{1}{2} (△_{W_{0}(𝔎)}f) (β_{t}) dt.

Here △_{W_{0}(𝔎)} is the Laplace-Beltrami operator defined in Definition 3.4, where (𝔎, 0) is viewed as a Lie group.

Proof. Use the proof of Lemma 2.16 with β, G in place of χ, G_{0}.

Lemma 3.9 (Semimartingale properties of h_{t·}). Let b be a 𝔎-valued Brownian Sheet (see Definition 3.5)Let h_{ts} be the solution to

h_{δts} = h_{ts}b_{δts} with h_{0s} = e. (3.2)

Then the process s ↦ h_{ts} is a K-valued Brownian motion with parameter t. Furthermore one can choose a version of h which is jointly continuous in both parameters s and t. In future, h will be taken to be this jointly continuous solution. Note:- Eq. [3.2] is to be interpreted like Eq. [2.4].

Proof. Let s_{i} = i/n. Then {0 = s_{0} < s_{1} < ⋯ < s_{n} = 1} is a partition of [0, T]. For convenience, let △_{i}b (t) ≡ b_{ts_{i}} − b_{ts_{i−1}}. We compute

δ_{t} (h_{ts_{i}}h_{ts_{i−1}}^{−1})

= h_{ts_{i}}b_{δts_{i}}h_{ts_{i−1}}^{−1} − h_{ts_{i}}b_{δts_{i−1}}h_{ts_{i−1}}^{−1}

= h_{ts_{i}}△_{i}b (δt) h_{ts_{i−1}}^{−1}

= (h_{ts_{i}}h_{ts_{i−1}}^{−1}) Ad_{h_{ts_{i−1}}} △_{i}b (δt)

= (h_{ts_{i}}h_{ts_{i−1}}^{−1}) Ad_{h_{ts_{i−1}}} △_{i}b (dt) + \frac{1}{2}d_{t} (h_{ts_{i}}h_{ts_{i−1}}^{−1}) Ad_{h_{ts_{i−1}}} △_{i}b (dt)

+ \frac{1}{2}Ad_{h_{ts_{i−1}}} [b_{dts_{i−1}}, △_{i}b (dt)]

= (h_{ts_{i}}h_{ts_{i−1}}^{−1}) Ad_{h_{ts_{i−1}}} △_{i}b (dt) + \frac{1}{2}d_{t} (h_{ts_{i}}h_{ts_{i−1}}^{−1}) Ad_{h_{ts_{i−1}}} △_{i}b (dt)

= (h_{ts_{i}}h_{ts_{i−1}}^{−1}) δ_{t} ∫_{0}^{t} Ad_{h_{τs_{i−1}}} △_{i}b (dτ) ,

where we have used that fact that b_{ts_{i−1}} ∈ 𝔉_{1s_{i−1}} and that △_{i}b (·) is independent of 𝔉_{1s_{i−1}}. Thus

δ_{t} (h_{ts_{i}}h_{ts_{i−1}}^{−1}) = (h_{ts_{i}}h_{ts_{i−1}}^{−1}) δ_{t} ∫_{0}^{t} Ad_{h_{τs_{i−1}}} △_{i}b (dτ) with h_{0s_{i}}h_{0s_{i−1}}^{−1} = e.

It suffices to show that {∫_{0}^{·} Ad_{h_{ts_{i−1}}} △_{i}b (dt)}_{i∈{1,··· ,n}} is a 𝔎^{n}-valued Brownian

motion with parameter 1/n, since this will imply that t → {h_{ts_{i}}h_{ts_{i−1}}^{−1}}_{i∈{1,··· ,n}} is

a K^{n}-valued Brownian motion with the same parameter. But this is true by Levy’s criterion and the following computation of quadratic variations.

Let J_{t} denote the joint quadratic variation

∫_{0}^{t} Ad_{h_{τs_{i−1}}} △_{i}b (dτ) Ad_{h_{τs_{j−1}}} △_{j}b (dτ) .

Then

dJ_{t} = Ad_{h_{ts_{i−1}}} △_{i}b (dt) Ad_{h_{ts_{j−1}}} △_{j}b (dt)
= ∑_{A,B} (Ad_{h_{ts_{i−1}}} A ⊗ Ad_{h_{ts_{j−1}}} B) △_{i}b^{A} (dt) △_{j}b^{B} (dt)

= δ_{ij}△_{i}sdt ∑_{A} (Ad_{h_{ts_{i−1}}} A)^{⊗2}

= \frac{δ_{ij}}{n} ∑_{A} (Ad_{h_{ts_{i−1}}} A)^{⊗2} dt

= \frac{δ_{ij}}{n} ∑_{A} A^{⊗2}dt.

Thus, in particular, Law (h_{T s_{1}}h_{T s_{0}}^{−1}, ⋯, h_{T s_{n}}h_{T s_{n−1}}^{−1}) is Heat Kernel Measure on K^{n} at time T/n. But this implies that s → h_{T ·} is a K-valued Brownian motion with parameter T.

We have only to show that h_{ts} satisfies the hypothesis of Theorem 8.2. That is we must show that

P [d (h_{ts}, h_{τσ})^{p}] ≤ C [(t − τ)^{2} + (s − σ)^{2}]^{\frac{m+β}{2}} .

The proof is essentially the same as that done in Theorem 3.8 of Driver [11] with the modification that G (s.σ) is used in place of G_{0} (s, σ). in particular, see Eq. [3.12] of [11].

Theorem 3.10 (Brownian motion exists on W_{e} (K)). Let h be the jointly continuous solution of Eq. [3.2]. Let h_{t} denote the element s → h_{ts} in W_{e} (K). Let k be an element of W_{e} (K). Then t → kh_{t} is a W_{e} (K)-valued Brownian motion starting from the path k.

Proof. Let

f (σ) ≡ F (σ_{s_{1}}, ⋯, σ_{s_{n}}) ,

be a smooth cylindrical function where ℙ is the partition {0 < s_{1} < ⋯ < s_{n} < 1}. Let A^{(i)}, π_{ℙ} and L_{ℙ} ≡ ∑ G (s_{i}, s_{j}) A^{(i)}A^{(j)} be as in Lemma 3.7. Let h^{ℙ} (t) ≡ (π_{ℙ} ∘ h_{t}) and let k^{ℙ} = π_{ℙ} ∘k. Let ℓ_{k^{ℙ}} denote left translation by the element k^{ℙ} ∈ K^{n} and let ℓ_{k} be left translation by the path k ∈ W_{e} (K). Simplifying, we get

df (kh_{t}) = d (f ∘ ℓ_{k}) (h_{t}) = d (F ∘ π_{ℙ} ∘ ℓ_{k}) (h_{t}) = d (F ∘ ℓ_{k^{ℙ}}) (h^{ℙ} (t)) .

By Eq. [3.2] the K^{n}-valued process h^{ℙ} satisfies

h^{ℙ} (δt) = h^{ℙ} (t) b^{ℙ} (δt) with h^{ℙ} (0) = e,

where e denotes the identity element in K^{n}. Then by Ito’s Lemma, we have

df (kh_{t}) = ∑_{A,i} (A^{(i)}F ∘ ℓ_{k^{ℙ}}) (h^{ℙ} (t)) b^{A} (δt, s_{i})
= ∑_{A,i} (A^{(i)}F ) ∘ ℓ_{k^{ℙ}} (h^{ℙ} (t)) b^{A} (dt, s_{i})

+\frac{1}{2} ∑_{A,i} d_{t} [(A^{(i)}F ) ∘ ℓ_{k^{ℙ}} (h^{ℙ} (t))] b^{A} (dt, s_{i}) . (3.3)

The quadratic variation

d_{t}[ (A^{(i)}F ) ∘ ℓ_{k^{ℙ}} (h^{ℙ} (t))]b^{A} (dt, s_{i})

= ∑_{B,j} (B^{(j)}A^{(i)}F ) ∘ ℓ_{k^{ℙ}} (h^{ℙ} (t)) b^{B} (dt, s_{j}) b^{A} (dt, s_{i})

= ∑_{j} (A^{(j)}A^{(i)}F ) ∘ ℓ_{k^{ℙ}} (h^{ℙ} (t)) G (s_{i}, s_{j}) dt.

Here we have used the fact (see Lemma 8.3) that the quadratic variation

b^{B} (dt, s_{j}) b^{A} (dt, s_{i}) = ⟨A, B⟩_{𝔎} G (s_{i}, s_{j}) dt.

Returning to Eq. [3.3] yields

df (kh_{t}) = dmartingale + \frac{1}{2} ∑_{A,i,j} G (s_{i}, s_{j}) (A^{(i)}A^{(j)}F ) (k^{ℙ}h^{ℙ} (t)) dt.

Invoking Lemma 3.7 yields

df (kh_{t}) = dmartingale + \frac{1}{2} (△_{W_{e}(K)}f) (kh_{t}) dt

for any smooth cylinder function f. Thus t → h (t, ·) is a Brownian motion on L (K).

Definition 3.11 (Heat Kernel measure on W_{e} (K)). Let k be an element of W_{e} (K). Let t → h_{t} be a W_{e} (K)-valued Brownian motion so that h_{0} = k a.s. Then, as in the finite-dimensional manifold case, Heat Kernel measure ν_{T}^{W_{e}(K)} (k, dγ) is defined to be the law of h (T, ·).

Remark 3.12 (Heat Kernel measures solve the Heat Equation). Let ℙ be the partition {0 < s_{1} < ⋯ < s_{n} < 1}. Let A^{(i)}, π_{ℙ} and L_{ℙ} be as in Lemma 3.7. Let f ≡ F ∘ π_{ℙ} be a smooth cylinder function for some F ∈ C^{∞} (K^{n}). Let

u (t, k) ≡ ∫ f (γ) ν_{t}^{W_{e}(K)} (k, dγ)

Let h be a W_{e} (K)-valued Brownian motion starting from k ∈ W_{e} (K). Then ν_{t}^{W_{e}(K)} (k, dγ) is the law of h. Let G^{−1} be the n × n matrix that is inverse to (G (s_{i}, s_{j})). Endow K^{n} with the metric ⟨A^{(i)}, B^{(j)}⟩ = ⟨A, B⟩_{𝔎} G_{kj}^{−1} so that the Laplacian on K^{n} (viewed as a Riemannian manifold) is the operator L_{ℙ} = ∑_{i,j,A} G (s_{i}, s_{j}) A^{(i)}A^{(j)}. Now t → h^{ℙ} (t) ≡ π_{ℙ} ∘ h_{t} satisfies the martingale characterization of a Brownian motion on K^{n} with this metric since by Lemma 3.7

F ∘ h^{ℙ} (dt) = F ∘ π_{ℙ} ∘ h_{dt} = dmartingale + \frac{1}{2} (L_{ℙ}F) ∘ h^{ℙ} (t) dt.

Thus

u (t, k) = Ef ∘ h_{t} = EF ∘ h^{ℙ} (t) = (exp (\frac{t}{2}L_{ℙ}) F ) ∘ π_{ℙ} (k) . (3.4)

By Lemma 3.7

(L_{ℙ}^{n}F) ∘ π_{ℙ} = △_{W_{e}(K)} (L_{ℙ}^{n−1}F ∘ π_{ℙ}) = △_{W_{e}(K)}^{n} (F ∘ π_{ℙ}) .

So in particular,

(exp (\frac{t}{2}L_{ℙ}) F ) ∘ π_{ℙ} = ∑_{ℕ} \frac{t^{n}}{2^{n}} (L_{ℙ}^{n}F) ∘ π_{ℙ}

= ∑_{ℕ} \frac{t^{n}}{2^{n}} △_{W_{e}(K)}^{n} (F ∘ π_{ℙ})

= exp (\frac{t}{2}△_{W_{e}(K)}) (F ∘ π_{ℙ}) .

Returning to Eq. [3.4] yields

u (t, k) = (exp (\frac{t}{2}△_{W_{e}(K)}) f) (k) .

Corollary 3.13. Heat Kernel measure ν_{T}^{W_{e}(K)} (e, dγ) and Wiener measure with parameter t are the same measure.

Proof. By Definition 3.11, Heat Kernel measure is the law of h_{t·}. By Lemma 3.9 s → h_{ts} is a Brownian motion on K with parameter t. Thus heat kernel measure and Wiener measure are the same.

3.2. Semimartingale Properties of X_{ts}: Let X_{ts} be as in Theorem 2.25. Then X is a Brownian bridge sheet on 𝔎. Brownian Sheets are easier to work with than Brownian bridge Sheets (they are martingales in both their parameters for instance). The goal of this section is to write X_{t·} as a linear functional of b_{t·} ,a Brownian sheet. To motivate this decomposition we first introduce Proposition 3.14. The Brownian bridge X̃_{·} is supposed to play the role of X_{t·} but with one fewer parameter.

Proposition 3.14. Let X̃ be the canonical process on C ([0, 1] → ℝ). That is X̃_{s} sends a path γ to its evaluation γ (s) at time s. Let the

P_{t}^{ℝ}(x) ≡ \frac{1}{\sqrt{2πt}} exp (−\frac{x^{2}}{2t} )

be Heat Kernels on ℝ. Define a new process b̃ by setting

b̃_{s} ≡ X̃_{s} − ∫_{0}^{s} ∇ ln P_{1−σ}^{ℝ}(X̃_{σ})dσ = X̃_{s} + ∫_{0}^{s} \frac{X̃_{σ}}{1 − σ} dσ.

Then b̃ is a standard ℝ-valued Brownian motion.

Notation 3.15. Let μ_{ℝ} be Wiener Measure on C ([0, 1] → ℝ). Let μ_{0}^{ℝ} = Law X̃ be pinned Wiener measure on ℝ (μ_{0}^{ℝ} is the measure μ_{0} in Definition 2.9 if K is taken to be ℝ). Let 𝔉_{t} be the σ-algebra generated by the X̃_{t} with t ∈ [0, s]. Let Z_{s} ≡ d (μ_{0}^{ℝ} ↓ 𝔉_{s}) /d (μ_{ℝ} ↓ 𝔉_{s}).

Proof. Now Z_{s} = P_{1−s}^{ℝ} \binom{}{X_{s}} /P_{1}^{ℝ} (0). By definition, X̃_{·} is a Brownian motion with respect to the measure μ_{ℝ}. Hence by The Meyer-Girsanov Theorem , which we state as Theorem 3.16 below for convenience,

N_{·} ≡ X̃_{·} − ∫_{0}^{·} \frac{dX̃_{s}dZ_{s}}{Z_{s}}

is a local martingale. This expression has the same quadratic variation as X̃_{·} (since the measures μ_{ℝ} and μ_{0}^{ℝ} are equivalent on 𝔉_{s} when s < 1). Thus this expression N_{·} is a Brownian motion by Levy’s criterion. Computing directly, we see that

dX̃_{s}dZ_{s} = dX̃_{s}d_{s} exp [log P_{1−s}^{ℝ} \binom{}{X_{s}} − log P_{1}^{ℝ} (0)]
= exp [log P_{1−s}^{ℝ} \binom{}{X_{s}} − log P_{1}^{ℝ} (0)] (∇ log P_{1−s}^{ℝ}) \binom{}{X_{s}} dX̃_{s}dX̃_{s}
= Z_{s} (∇ log P_{1−s}^{ℝ}) \binom{}{X_{s}} ds

Thus N_{·} = X̃_{·} − ∫_{0}^{·} (∇ log P_{1−s}^{ℝ}) \binom{}{X_{s}} ds = b̃_{·} and we are done.

Theorem 3.16 (Meyer-Girsanov, see [30]). Let P and Q be equivalent measures and let Z_{s} ≡ E [dQ/dP |𝔉_{s}]. Let X̃ be a semimartingale under P with decomposition M + A (where M is a local martingale and A has finite variation). Then X̃ is also a semimartingale under Q with decomposition N + C where

N_{·} = X̃_{·} − ∫_{0}^{·} \frac{dX̃_{s}dZ_{s}}{Z_{s}}

is a Q-local martingale and C ≡ X̃ − N is a finite variation process.

Definition 3.17. Define the following linear maps:- 1. Define continuous 𝔎-valued linear maps on paths,

T_{t}, S_{t} : C ([0, 1] → 𝔎) → 𝔎,

by setting

T_{t}(ω) ≡ ω(t) − ∫_{0}^{t} ω(τ)\frac{(1 − t)}{(1 − τ)^{2}} dτ if t ∈ [0, 1) .
S_{t}(ω) ≡ ω(t) + ∫_{0}^{t} \frac{ω(τ)}{(1 − τ)}dτ if t ∈ [0, 1) .

2. Let U_{1} and U_{2} be the subsets of C ([0, 1] → 𝔎) on which the limits lim_{t→1} T_{t}(ω) and lim_{t→1} S_{t}(ω) exist respectively. Then define maps T_{1} and S_{1} from C ([0, 1] → 𝔎) to 𝔎 by setting

T_{1} ≡ 1_{U_{1}} (ω) lim_{t→1} T_{t}(ω).
S_{1} ≡ 1_{U_{2}} (ω) lim_{t→1} S_{t}(ω).

Remark 3.18. Notice that in Proposition 3.14 we wrote the underlying Brownian motion b̃_{·} as S \binom{}{X_{·}} (·). Similarly we shall prove the process b_{t·} ≡ S (X_{t·}) is a Brownian Sheet and that X_{t·} can be written as T (b_{t·}).

Theorem 3.19 (Decomposition of the Brownian bridge sheet). Let X be the Brownian bridge sheet from Theorem 2.25. Define b by setting

b_{ts} ≡ S_{s}(X_{t·}) = X_{ts} + ∫_{0}^{s} \frac{X_{tσ}dσ}{1 − σ} for any t, s ∈ [0, 1] .

Then b is a Brownian sheet on 𝔎 and X_{ts} can be recovered from b as:

X_{ts} = T_{s}(b_{t·}) = b_{ts} − ∫_{0}^{s} b_{tσ} \frac{(1 − s)}{(1 − σ)^{2}} dσ. (3.5)

We shall defer the proof of Theorem 3.19 until after the Lemma 3.21 below.

Remark 3.20. For another explicit computational proof of this Theorem, see Theorem 8.5 in the Appendix 8.

Lemma 3.21 (Properties of the transformations S and T). Define a map T from H (𝔎) to H_{0} (𝔎) by setting T (ω) (t) = T_{t} (ω). Define a map S from H_{0} (𝔎) to H (𝔎) by setting S (ω) (t) = S_{t} (ω). Then:-

1. S is well-defined and is a unitary isomorphism from H_{0} (𝔎) to H (𝔎).

2. T is well-defined and is the inverse of S.

Proof. Let ω ∈ H (𝔎). By an integration-by-parts we can express T more concisely as

T (ω)(t) = ω(t) − ∫_{0}^{t} ω(τ)\frac{(1 − t)}{(1 − τ)^{2}} dτ
= ω(t) − (1 − t) ∫_{τ=0}^{τ=t} ω(τ)d\frac{1}{1 − τ}
= ∫_{0}^{t} \frac{(1 − t)}{1 − τ}ω′(τ)dτ.

Thus we have the inequality

|T (ω)(t)| ≤ (1 − t) ∫_{0}^{t} \frac{1}{1 − τ} |ω′(τ)| dτ

≤ |ω|_{H(𝔎)} (1 − t) \sqrt{∫_{0}^{t} \frac{dτ}{(1 − τ)^{2}}}
= |ω|_{H(𝔎)} \sqrt{t (1 − t)}
→ 0 as t → 1.

Thus T_{t} is continuous on H (𝔎), and Im T ⊂ L (𝔎).

Let 𝔘_{0} denote the subspace of functions of the form σ = ∫_{0}^{·} x (t) dt where x is in the continuous maps from [0, 1] to 𝔎 so that its average, ∫_{0}^{1} x (t) dt, is 0. By the Stone-Weierstrass Theorem, C ([0, 1] → 𝔎) is dense in L^{2} ([0, 1] → 𝔎, dλ) and in particular continuous functions with zero average are dense in the space of L^{2} functions with zero average. Thus by the isometry provided by the map x → ∫ xdt from L^{2} ([0, 1] → 𝔎, dλ) to H (𝔎) we see that 𝔘_{0} is dense in H_{0} (𝔎) in the H_{0} (𝔎) norm topology.

We claim S is a norm-preserving map from 𝔘_{0} to H (𝔎). Let σ = ∫_{0}^{·} x (t) dt in 𝔘_{0}. Computing, we see that

|S(σ)|_{H(𝔎)}^{2} = ∫_{0}^{1} |σ′(t) + \frac{σ(t)}{(1 − t)}|_{𝔎}^{2} dt
= |σ|_{H_{0}(𝔎)}^{2} + 2 ∫_{0}^{1} ⟨σ′(t), \frac{σ(t)}{(1 − t)}⟩_{𝔎} dt + ∫_{0}^{1} |σ(t)|_{𝔎}^{2} \frac{dt}{(1 − t)^{2}}
= |σ|_{H_{0}(𝔎)}^{2} + 2 ∫_{0}^{1} ⟨σ′(t), \frac{σ(t)}{(1 − t)}⟩_{𝔎} dt + ∫_{t=0}^{t=1} |σ(t)|_{𝔎}^{2} d (\frac{1}{1 − t})

= |σ|_{H_{0}(𝔎)}^{2} + lim_{t→1} \frac{|σ(t)|_{𝔎}^{2}}{1 − t}
= |σ|_{H_{0}(𝔎)}^{2} − 2 ⟨σ(1), x (1)⟩_{𝔎}
= |σ|_{H_{0}(𝔎)}^{2} since σ (1) = 0.

By the Bounded Limit Theorem, we can extend S to a map \overline{S} on all of H_{0} (𝔎) by defining

\overline{S} (ω) ≡ lim_{n→∞} S (ω_{n}) for any ω_{n} ∈ 𝔘_{0}, ω_{n} → ω in H_{0} (𝔎) .

Although \overline{S} and S agree on 𝔘_{0} they could be different maps on H_{0} (𝔎). We will check that this is not the case. Notice that the evaluation map sending ω in H (𝔎) to ω (t) in 𝔎 is a bounded linear map. Also if s < 1, the map S_{s}|_{H(𝔎)} is a continuous map from H (𝔎) to 𝔎 in the H (𝔎)-norm. Therefore, if ω_{n} in 𝔘_{0} converges to ω in the H (𝔎)-norm, we have for all s < 1,

\overline{S} (ω) (s) = lim_{n→∞} S (ω_{n}) (s) = lim_{n→∞} S_{s} (ω_{n}) = S (ω) (s) .

Thus S = \overline{S} which is a norm-preserving map from H_{0} (𝔎) into H (𝔎).

Let x in H_{0} (𝔎). Let y = S (x) in H (𝔎) and z = T (y) in L (𝔎) so that z = T ∘ S (x). As before, x (0) = z (0) = 0. Letting t < 1 and computing, we have

z′ (t) = y′ (t) − ∫_{0}^{t} \frac{y′ (τ) dτ}{1 − τ}
= x′ (t) + \frac{x (t)}{1 − t} − ∫_{0}^{t} \frac{x′ (τ) dτ}{1 − τ} − ∫_{0}^{t} \frac{x (τ) dτ}{(1 − τ)^{2}}

= x′ (t) + \frac{x (t)}{1 − t} − ∫_{0}^{t} \frac{dx (τ)}{1 − τ} − ∫_{0}^{t} \frac{x (τ) dτ}{(1 − τ)^{2}}
= x′ (t) .

So T ∘ S is the identity on H_{0} (𝔎) and so T is a surjective norm-preserving from Im S to H_{0} (𝔎).

Let x in H (𝔎), y = T (x) in L (𝔎) and z = S_{t} (y) so that z_{t} = S_{t} ∘ T (x). Since x (0) = z_{0} = 0 and

\frac{d}{dt}z_{t} = y′ (t) + \frac{y (t)}{1 − t}
= x′ (t) − ∫_{0}^{t} \frac{x′ (τ) dτ}{1 − τ} + \frac{y (t)}{1 − t}
= x′ (t) − \frac{T (x) (t)}{1 − t} + \frac{y (t)}{1 − t}
= x′ (t) .

So S_{t} ∘ T (x) = x (t) for any x in H (𝔎).

Now we show that T maps a dense subspace 𝔘 of H (𝔎) into H_{0} (𝔎). If x ∈ 𝔘, then

S ∘ T (x) (t) = S_{t} ∘ T (x) = x (t) .

In particular, x belongs to Im S. Thus Im S contains a dense set 𝔘 and so is all of H (𝔎). So S will be a unitary isomorphism between H_{0} (𝔎) and H (𝔎), T will be its inverse, and we shall be done.

Let 𝔘 denote the subspace of functions of the form σ = ∫_{0}^{·} x (t) dt where x is in C ([0, 1] → 𝔎). By the Stone-Weierstrass Theorem, C ([0, 1] → 𝔎) is dense in L^{2} ([0, 1] → 𝔎, dλ). Thus by the isometry provided by the map x → ∫ xdt from L^{2} ([0, 1] → 𝔎, dλ) to H (𝔎) we see that 𝔘 is dense in H (𝔎) in the H (𝔎) norm topology. Then

T (σ)′ (t) = σ′ (t) − ∫_{0}^{t} \frac{σ′(τ)}{1 − τ} dτ = x (t) − ∫_{0}^{t} \frac{x(τ)}{1 − τ} dτ.

∫_{0}^{1} |T (σ)′ (t)|^{2} dt

= ∫_{0}^{1} |x (t) − ∫_{0}^{t} \frac{x (τ)}{1 − τ} dτ|_{𝔎}^{2} dt

< 2 sup_{[0,1]} |x (t)| ∫_{0}^{1} 1 + |∫_{0}^{t} \frac{1}{1 − τ} dτ|^{2} dt

= 2 ∥x∥_{∞} ∫_{0}^{1} 1 + |log (1 − t)|^{2} dt

= 2 ∥x∥_{∞} ∫_{0}^{1} 1 + |log t|^{2} dt.

Letting t = −u^{2}/2 we have

|T (σ)|_{H_{0}(𝔎)}^{2} ≤ 2 sup_{[0,1]} |σ′ (t)| ∫_{0}^{∞} u (1 + \frac{u^{4}}{4} ) exp (−u^{2}/2) du < ∞.

Proof. of Theorem 3.19:

First we show that

E ⟨b_{ts}, A⟩_{𝔎} ⟨b_{τσ}, B⟩_{𝔎} = (t ∧ τ) G (s, σ) ⟨B, A⟩_{𝔎}

Recall b_{ts}^{A} ≡ ⟨b_{ts}, A⟩_{𝔎} and X_{ts}^{A} ≡ ⟨X_{ts}, A⟩_{𝔎}. Let

l_{s} (x) ≡ ∫_{0}^{1} α_{s} (du) x (u)

where

α (du) = [δ_{s} (u) + 1_{[0,s]} \frac{1}{1 − u}] du

is a positive measure on [0, 1]. Then

l_{s} (x) = x (s) + ∫_{0}^{1} x (u) \frac{du}{1 − u} = S_{s} (x) .

Define b_{ts} ≡ S_{s} (X_{t·}) as in Definition 3.17. So

Eb_{ts}^{A}b_{τσ}^{B} = E ∫ α_{s} (du) α_{σ} (dν) X_{tu}^{A}X_{τν}^{B}. (3.6)

By Tonelli’s Theorem and Hölder’s inequality, we have

E ∫ α_{s} (du) α_{σ} (dν) |X_{tu}^{A}X_{τν}^{B} | ≤ ∫ α_{s} (du) α_{σ} (dν) \sqrt{E (X_{tu}^{A})^{2} E (X_{τν}^{B})^{2}}
= ∫ α_{s} (du) α_{σ} (dν) \sqrt{tτG_{0} (u, u) G_{0} (ν, ν)} < ∞.

Thus applying Fubini to Eq. [3.6] we see that

Eb_{ts}^{A}b_{τσ}^{B} = ∫ α_{s} (du) α_{σ} (dν) EX_{tu}^{A}X_{τν}^{B}
(3.7)  = (t ∧ τ) ⟨A, B⟩_{𝔎} ∫ α_{s} (du) ∫ α_{σ} (dν) G_{0} (u, ν) .

Let h run through an orthonormal basis of H_{0} (𝔎). Then

G_{0} (u, ν) = ⟨G_{0} (u, ·) , G_{0} (ν, ·)⟩_{H_{0}(𝔎)}
= ∑ ⟨G_{0} (u, ·) , h⟩_{H_{0}} ⟨G_{0} (u, ·) , h⟩_{H_{0}(𝔎)}
= ∑ h (u) h (ν) .

Returning to Eq. [3.7] we get

E ⟨b_{ts}, A⟩_{𝔎} ⟨b_{τσ}, B⟩_{𝔎} = (t ∧ τ) ⟨A, B⟩_{𝔎} ∑ ∫ h (u) α_{s} (du) ∫ h (ν) α_{σ} (dν)
(3.8)  = (t ∧ τ) ⟨A, B⟩_{𝔎} ∑ S_{s} (h) S_{σ} (h) .

Let U ≡ {x|_{[0,1]} : x ∈ C^{∞} (ℝ)}. The map S : H_{0} (𝔎) → H (𝔎) is a unitary isomorphism by the previous Lemma 3.21 and so the S (h) run through an orthonormal basis of H (𝔎). Exploiting this fact,

E ⟨b_{ts}, A⟩_{𝔎} ⟨b_{τσ}, B⟩_{𝔎}
= (t ∧ τ) ⟨A, B⟩_{𝔎} ∑ S (h) (s) S (h) (σ)
= (t ∧ τ) ⟨A, B⟩_{𝔎} ∑ ⟨G (s, ·) , S (h)⟩_{H(𝔎)} ⟨G (σ, ·) , S (h)⟩_{H(𝔎)}
= (t ∧ τ) ⟨A, B⟩_{𝔎} ⟨G (s, ·) , G (σ, ·)⟩_{H(𝔎)}
= (t ∧ τ) ⟨A, B⟩_{𝔎} G (s, σ) .

Thus b is a 𝔎-valued Brownian sheet.

It remains to show that T (b_{t·}) (s) = X_{ts}. Define H^{ε} (𝔎) which is to be thought of as “H (𝔎) |_{[0,1−ε]}” as follows:- Given a function h : [0, 1 − ε] → 𝔎 such that h (0) = 0, define (h, h)_{H^{ε}(𝔎)} = ∞ if h is not absolutely continuous and set (h, h)_{H^{ε}(𝔎)} =

∫_{0}^{1−ε} |h′ (s)|^{2} ds otherwise. Define

H^{ε} (𝔎) ≡ {h : [0, 1 − ε] → 𝔎|h (0) = 0 and (h, h)_{H^{ε}(𝔎)} < ∞} .

H^{ε} is dense in W_{0}^{ε}, where

W_{0}^{ε} (𝔎) ≡ {σ ∈ C ([0, 1 − ε] → 𝔎) |σ (0) = 0}

is equipped with the sup-norm topology. Define bounded linear transformations T^{ε} and S^{ε} on W_{0}^{ε} (𝔎) by requiring

T^{ε}(x)(t) ≡ x(t) − ∫_{0}^{t} x(τ)\frac{(1 − t)}{(1 − τ)^{2}} dτ;
S^{ε}(x)(t) ≡ x(t) + ∫_{0}^{t} \frac{x(τ)}{(1 − τ)}dτ.

Now for any h ∈ H (𝔎) or H_{0} (𝔎), h|_{[0,1−ε]} ∈ H^{ε} (𝔎). Also h ∈ H (𝔎) implies that (T (h)) |_{[0,1−ε]} = T^{ε} (h|_{[0,1−ε]}). Furthermore h ∈ H_{0} (𝔎) implies that (S (h)) |_{[0,1−ε]} = S^{ε} (h|_{[0,1−ε]}). For any x ∈ H^{ε} there is some h ∈ H_{0} (𝔎) so that h|_{[0,1−ε]} = x. Using this fact and the fact that T = S^{−1} from Lemma 3.21, we see that for any x ∈ H^{ε} (𝔎)

T^{ε} ∘ S^{ε} (x) = T^{ε} ∘ (S (h) |_{[0,1−ε]}) = h|_{[0,1−ε]} = x.

By continuity, we have T^{ε} ∘ S^{ε} (x) = x for any x ∈ W_{0}^{ε}. Thus for any s < 1 − ε, we have

T (b_{t·}) (s) = T^{ε} (b_{t·}|_{[0,1−ε]}) (s) = T^{ε} ∘ S^{ε} (X_{t·}|_{[0,1−ε]}) (s) = X_{ts}.

Thus

X_{ts} = T (b_{t·}) (s) = b_{ts} − ∫_{0}^{s} b_{tσ} \frac{(1 − s)}{(1 − σ)^{2}} dσ,

which is exactly Eq. [3.5].

3.3. Abelian Loop group Examples.

3.3.1. The Simply-Connected Lie group (ℝ^{d}, +):

Lemma 3.22. On the Loop space of ℝ^{d}, Heat Kernel measure and pinned Wiener measure are the same.

Proof. Our Lie group here is K = (ℝ^{d}, +) with Lie algebra 𝔎 = ℝ^{d}. Our probability space is (C ([0, 1] → K) , Law Σ) as in Definition 2.22. Eq. (2.6) becomes

Σ (δt, s) = χ (δt, s) with Σ (0, s) = 0, ∀s ∈ [0, 1] .

In other words, Σ = χ. This implies that Heat Kernel measure on L (ℝ^{d}) equals Law χ (t, ·). But χ (t, ·) is a standard Brownian bridge from 0 to 0. Pinned Wiener measure is the law of this Brownian bridge. Hence in ℝ^{d}, Heat Kernel Measure and Pinned Wiener Measure are the same measure.

3.3.2. The Lie group S^{1} with fundamental group ℤ: Realize the Lie group S^{1} as {(cos 2πθ, sin 2πθ) : θ ∈ [0, 1]}, its imbedding in ℝ^{2}. Specify the left-invariant metric by setting |∂_{θ}| = 1. Let Heat Kernel measure ν_{T}^{S^{1}} (x, ·) be the family of measures in Definition 2.20. Let pinned Wiener measure μ_{0,T}^{S^{1}} be the measure on the loop space L (S^{1}) as in Definition 2.9. Let Wiener measure μ_{T}^{S^{1}}on W_{e} (S^{1}) be as in Definition 2.7. Let

W_{0} (ℝ) ≡ {C ([0, 1] → ℝ) : σ (0) = 0}

be the Wiener space on ℝ. Let π_{t} : L (S^{1}) → ℝ be as usual the evaluation map. By abuse of notation, let π_{t} : W_{e} (S^{1}) → ℝ be also be the evaluation map.

We show, in the K = S^{1} case, that Heat Kernel Measure is equivalent to Pinned Wiener Measure restricted to the null-homotopic loops. Thus Heat Kernel Measure is absolutely continuous with Pinned Wiener Measure. However, as mentioned in the Introduction, the two measures are not equivalent since S^{1} is not simply connected.

We shall need to explicitly compute the Heat Kernel Measure on S^{1} and this is provided for the reader’s convenience in the following Lemma:

Lemma 3.23 (Heat Kernel measure on S^{1}). Let ψ be the local chart from ℝ to S^{1} taking x ↦ (cos 2πx, sin 2πx).Then Heat Kernel Measure on S^{1} has the following representation:

P_{t}^{S^{1}} (ψ (x)) = ∑_{α∈ℤ} P_{t}^{ℝ} (x + α) . (3.9)

Proof. Since the right hand side of Eq. [3.9] is periodic there exist a unique map P (t, θ) from [0, ∞) × S^{1} → ℝ so that P (t, ψ (x)) = ∑_{α∈ℤ} P_{t}^{ℝ} (x + α).

∂_{x}P (t, ψ (x)) = [∂_{θ}P (t, ·)] (ψ (x)) .

Thus

(∂_{t} − \frac{1}{2}∂_{θ}^{2}) P (t, θ) |_{θ=ψ(x)} = (∂_{t} − \frac{1}{2}∂_{x}^{2}) P (t, ψ (x))
= ∑_{α∈ℤ} (∂_{t} − \frac{1}{2}∂_{x}^{2}) P_{t}^{ℝ} (x + α) = 0.

For any F ∈ C^{∞} (S^{1}), define a map u (t, θ) from [0, ∞) × S^{1} → ℝ by

u (t, θ) = ∫_{S^{1}} F (θ) P (t, θ^{−1}θ′) νol (dθ′) .

Take the support of F to be less than the entire circle. The appropriate local chart here is ψ restricted to some open interval (a, b) with |b − a| < 1. Let θ = ψ (y) for some y ∈ (a, b). Use this local chart and the fact that |∂_{x}|_{S^{1}} = 1, to get

u (t, ψ (y)) = ∫_{(a,b)} F (ψ (x)) P (t, ψ (x − y)) dx
= ∑_{α∈ℤ} ∫_{(a,b)} F (ψ (x)) P_{t}^{ℝ} (x − y + α) dx

= ∫_{(a,b)+α} F (ψ (x)) P_{t}^{ℝ} (x − y) dx
= ∫_{ℝ} F (ψ (x)) P_{t}^{ℝ} (x − y) dx.

So u (t, ψ (y)) → F (ψ (y)) as t → 0. Thus in this above sense, P (t, θ^{−1}θ′) → δ_{θ} as t → 0. Therefore P (t, θ) must be the Heat Kernel on S^{1}.

Remark 3.24. Recall from Definition 2.22 the following:- 1. Ω ≡ C ([0, 1] → L (S^{1})) .

2. Let Σ be the process from Theorem 2.17 so that Σ_{0} = e, where e denotes the identity loop.

3. P is defined to be Wiener Measure on C ([0, 1] → L (S^{1})). Explicitly, P ≡ Law Σ.

4. g_{ts} (x) ≡ x (t) (s), where x ∈ Ω, x (t) ∈ L (K), and x (t) (s) ∈ K.

5. By Theorem 2.19 we see that Law g_{t·} = ν_{t} (e, ·), the Heat Kernel measure on L (S^{1}) introduced in Definition 2.20.

6. 𝔉_{ts} ≡ σ ⟨g_{τσ} : τ ∈ [0, t] and σ ∈ [0, s]⟩ .

7. 𝔉 ≡ ∨_{(t,s)∈[0,1]^{2}}𝔉_{ts}.

Definition 3.25 (S^{1}-specific definitions). We will need the following:-

1. L_{0} (S^{1}) ≡ {σ ∈ L (S^{1}): σ is homotopic to e}; the null-homotopic loops in S^{1} based at e.

2. Abusing notation. let ψ also denote the map from W_{0} (ℝ) to W_{1} (S^{1}) taking the ℝ-valued path σ to the S^{1}-valued path (cos 2πσ, sin 2πσ).

3. ψ has a unique inverse ψ^{−1} : W_{1} (S^{1}) → W_{0} (ℝ) which is the unique lift of σ starting from 0 in ℝ.

Let μ_{T}^{ℝ} be Wiener measure on W_{0} (ℝ) with parameter T. Explicitly, use Definition 2.7 with K = ℝ and P_{t} (x^{−1}y) replaced by the Heat Kernel

P_{t}^{ℝ} (y − x) = \frac{1}{\sqrt{2πt}} exp \frac{(y − x)^{2}}{2t} .

Definition 3.26 (Wiener Measure conditioned on the integers). Let μ_{ℤ,T}^{ℝ} be the unique measure on W (ℝ) such that on simple functions f of the form

f (x_{·}) = F (x_{t_{1}}, ⋯, x_{t_{n}}) ,

where F ∈ C_{b}^{∞} (ℝ^{n}) and {0 = t_{0} < t_{1} < ⋯ < t_{n} < 1}, we have;

μ_{ℤ,T}^{ℝ} [f]

≡ \frac{1}{P_{T}^{S^{1}} (e)} ∫_{ℝ^{n}} F (x_{1}, ⋯, x_{n}) ∑_{α∈ℤ} P_{T (1−s_{n})}^{ℝ} (x_{n} + α) ∏_{i=1}^{n} P_{T △_{i}s}^{ℝ} (△_{i}x) dx_{i}

= \frac{1}{P_{T}^{S^{1}} (e)} ∫_{ℝ^{n}} F (x_{1}, ⋯, x_{n}) P_{T (1−t_{n})}^{S^{1}} (ψ (x_{n})) ∏_{i=1}^{n} P_{T △_{i}t}^{ℝ} (△_{i}x) dx_{i}

= \frac{1}{P_{T}^{S^{1}} (e)} ∫ μ_{T}^{ℝ} (dx) f (x) P_{T (1−t_{n})}^{S^{1}} (ψ (x_{t_{n}})) . (3.10)

We have yet to show the existence of such a measure. See remark 3.27 to see why μ_{ℤ,T}^{ℝ} deserves to be called “Wiener Measure conditioned on the integers with parameter T ”.

Remark 3.27 (Motivation for Definition 3.26). Our goal is to make explicit the heuristic definition

μ_{ℤ,T}^{ℝ} [f] = μ_{T}^{ℝ} [f (x) |x (1) ∈ ℤ] .

Take the function ∑_{α∈ℤ} P_{ε}^{ℝ} (π_{1} + α) which concentrates on paths which are near ℤ at time t = 1.We would like

μ_{ℤ,T}^{ℝ} [f] = lim_{ε↓0} μ_{T}^{ℝ} [f ∑_{α∈ℤ} P_{ε}^{ℝ} (π_{1} + α)] /μ_{T}^{ℝ} [∑_{α∈ℤ} P_{ε}^{ℝ} (π_{1} + α)]

to hold. Let F ∈ C_{b}^{∞} (ℝ^{n}) and

f (x) = F (x_{t_{1}}, ⋯, x_{t_{n}}) where {0 = t_{0} < ⋯ < t_{n+1} = 1} .

Then letting △_{i}t = t_{i} − t_{i−1} and △_{i}x = x_{i} − x_{i−1}, and using the fact that P_{ε}^{ℝ} (·) goes to the delta function at 0 as ε → 0; we should have

μ_{T,ℤ}^{ℝ} [f]

= lim_{ε↓0} \frac{∫_{ℝ^{n+1}} F (x_{1}, ⋯, x_{n}) ∑_{α∈ℤ} P_{ε}^{ℝ} (x_{n+1} + α) ∏_{i=1}^{n+1} P_{T △_{i}t}^{ℝ} (△_{i}x) dx_{i}}{∫_{ℝ} ∑_{α∈ℤ} P_{ε}^{ℝ} (x + α) P_{T}^{ℝ} (x) dx}

= \frac{∫_{ℝ^{n}} F (x_{1}, ⋯, x_{n}) ∑_{α∈ℤ} P_{T (1−t_{n})}^{ℝ} (x_{n} + α) ∏_{i=1}^{n} P_{T △_{i}t}^{ℝ} (△_{i}x) dx_{i}}{∑_{α∈ℤ} P_{T}^{ℝ} (α)} ,

where we have replaced x_{n+1} + α by x_{n+1} using the change-of-variables formula. Thus for simple functions, we should have

μ_{T,ℤ}^{ℝ} [f] = \frac{1}{∑_{α∈ℤ} P_{T}^{ℝ} (α)} ∫ f (x) ∑_{α∈ℤ} P_{T (1−t_{n})}^{ℝ} (x_{t_{n}} + α) μ_{T}^{ℝ} (dx) .

Now use Lemma 3.23 to see that ∑_{α∈ℤ} P_{T}^{ℝ} (α) = P_{T}^{S^{1}} (e) .

Theorem 3.28 (Heat Kernel and pinned Wiener measures on S^{1}). Let ψ be as in Definition 3.25. Let σ_{n} (s) ≡ ψ (ns), the minimum energy loop in the n^{th} homotopy class of S^{1}. Let ν_{T}^{S^{1}} (σ_{n}, ·) be as usual the Law of σ_{n} (·) g_{T ·}. Define a probability measure

ν̃_{T} ≡ ∑_{α∈ℤ} C_{α,T} ν_{T}^{S^{1}} (σ_{α}, ·) ,

where

C_{α,T} ≡ P_{T}^{ℝ} (0) exp (−\frac{1}{2T} α^{2}) P_{T}^{S^{1}} (e)^{−1} = P_{T}^{ℝ} (α) P_{T}^{S^{1}} (e)^{−1} .

Then Pinned Wiener Measure μ_{0,T}^{S^{1}} = ν̃_{T}. Exploiting the fact that the measures ν_{T}^{S^{1}} (σ_{α}, ·) live only on the α^{th} homotopy classes, we see that Heat Kernel Measure ν_{T}^{S^{1}} (e, ·) is equivalent to Pinned Wiener Measure restricted to the null-homotopic loops L_{0}. Furthermore, the Radon-Nikodym derivative

\frac{dμ_{0,T}^{S^{1}} ↓_{L_{0}}}{dν_{T}^{S^{1}} (e, ·)} = C_{0,T}

is a constant.

The proof of this Theorem will be deferred until we have some preliminary results.

Lemma 3.29 (The measure μ_{T,ℤ}^{ℝ}exists). Wiener Measure conditioned on the integers, μ_{ℤ,T}^{ℝ}, exists. Furthermore Pinned Wiener Measure on S^{1} pulls back to Wiener Measure conditioned on the integers. Explicitly, ψ_{∗}^{−1}μ_{0,T}^{S^{1}} = μ_{ℤ,T}^{ℝ} .

Proof. ψ from Definition 3.25 is a continuous bijection between the Wiener spaces W (R) and W (S^{1}). Let us compute ψ_{∗}^{−1}μ_{0,T}^{S^{1}} for simple functions. By Ito’s Lemma it is easily seen that ψ takes ℝ-valued Brownian motions with parameter T to S^{1}- valued Brownian motions with the same parameter. So let t → b_{t} be an ℝ-valued Brownian motion with parameter t. Then t → ψ (b_{t}) is an S^{1}-valued Brownian motion. Thus

∫ f (y) μ_{T}^{S^{1}} (dy) = ∫ f ∘ ψ (y) μ_{T}^{ℝ} (dy) .

Let F ∈ C^{∞} (S^{1} × ⋯ × S^{1}) and let

f (y) = F (y_{s_{1}}, ⋯, y_{s_{n}}) where {0 = s_{0} < ⋯ < s_{n} < 1} ,

for any path y ∈ W_{e} (S^{1}). Then

ψ_{∗}^{−1}μ_{0,T}^{S^{1}} [f] = μ_{0,T}^{S^{1}} [f ∘ ψ^{−1}]
= ∫ f ∘ ψ^{−1} (y) \frac{P_{t(1−s_{n})}^{S^{1}} (y_{s_{n}})}{P_{t}^{S^{1}} (e)}μ_{T}^{S^{1}} (dy)
= \frac{1}{P_{t}^{S^{1}} (e)} ∫ f (y) P_{t(1−s_{n})}^{S^{1}} (ψ (y_{s_{n}})) μ_{T}^{ℝ} (dy) .

This is precisely Eq. [3.10]. Thus the measure μ_{ℤ,T}^{ℝ} and μ_{0,T}^{S^{1}} is pulled back to it under the map ψ.

ON THE EQUIVALENCE OF MEASURES ON LOOP SPACE 29

Lemma 3.30. Let △_{i}u denote u_{i} − u_{i−1}. Then the functions

J_{1} (x_{1}, ⋯, x_{n}) ≡ P_{T (1−s_{n})}^{ℝ} (x_{n} − α) ∏_{i=1}^{n} P_{T △_{i}s}^{ℝ} (△_{i}x) ,

and

J_{2} (x_{1}, ⋯, x_{n}) = exp (\frac{−α^{2}}{2T} ) P_{T (1−s_{n})}^{ℝ} (x_{n} − αs_{n}) ∏_{i=1}^{n} P_{T △_{i}s}^{ℝ} (△_{i}x − α△_{i}s)

are the same. (i.e. J_{1} = J_{2}). Proof. We shall use the fact that

P_{t}^{ℝ} (x) = \frac{1}{\sqrt{2πt}} exp −\frac{x^{2}}{2t} .

Letting △_{n+1}s denote 1 − s_{n}, we have

J_{2} (x_{1}, ⋯, x_{n}) = exp (−α^{2}/2T ) P_{T (1−s_{n})}^{ℝ} (x_{n} − αs_{n}) ∏_{i=1}^{n} P_{T △_{i}s}^{ℝ} (△_{i}x − α△_{i}s) .

Let

I = −2T (log J_{2} + ∑_{i=1}^{n+1} log \sqrt{2πT △_{i}s}) .

Then

I = α^{2} + \frac{(x_{n} − αs_{n})^{2}}{1 − s_{n}} + ∑_{i=1}^{n} \frac{(△_{i}x − α△_{i}s)^{2}}{△_{i}s}
= α^{2} + \frac{x_{n}^{2} − 2αx_{n}s_{n} + α^{2}s_{n}^{2}}{1 − s_{n}} + ∑_{i=1}^{n} [\frac{△_{i}x^{2}}{△_{i}s} − 2α△_{i}x + α^{2}△_{i}s]
= α^{2} + \frac{x_{n}^{2} − 2αx_{n}s_{n} + α^{2}s_{n}^{2}}{1 − s_{n}} − 2αx_{n} + α^{2}s_{n} + ∑_{i=1}^{n} \frac{△_{i}x^{2}}{△_{i}s}
= \frac{x_{n}^{2} + α^{2}s_{n}^{2}}{1 − s_{n}} − 2αx_{n} (\frac{s_{n}}{1 − s_{n}} + 1) + α^{2} (1 + s_{n}) + ∑_{i=1}^{n} \frac{△_{i}x^{2}}{△_{i}s}
= \frac{x_{n}^{2} + α^{2}s_{n}^{2} − 2αx_{n} + α^{2} (1 − s_{n}^{2})}{1 − s_{n}} + ∑_{i=1}^{n} \frac{△_{i}x^{2}}{△_{i}s}
= \frac{(x_{n} − α)^{2}}{1 − s_{n}} + ∑_{i=1}^{n} \frac{△_{i}x^{2}}{△_{i}s}
= −2T (log J_{1} + ∑_{i=1}^{n+1} log \sqrt{2πT △_{i}s}) .

Hence we are done. Proof. of Theorem 3.28

Let the map ψ be as in Definition 3.25. It will suffice to show ψ_{∗}^{−1}μ_{0,T}^{S^{1}} is equivalent to ψ_{∗}^{−1}ν̃_{T}. If this is the case then for any measurable A ⊂ L (S^{1}) we have

A is a ν̃_{T} -null set
⟺ ν̃_{T} (e, 1_{A} ∘ ψ ∘ ψ^{−1}) = 0
⟺ ψ_{∗}^{−1}ν̃_{T} (e, 1_{ψ}^{−1}_{(A)}) = 0
⟺ ψ_{∗}^{−1}μ_{0,T}^{S^{1}} (1_{ψ}^{−1}_{(A)}) = 0
⟺ A is a μ_{0,T}^{S^{1}} -null set.

Thus we would be done by the Radon-Nikodym Theorem. The rest of the proof is devoted to computing ψ_{∗}^{−1}μ_{0,T}^{S^{1}} and ψ_{∗}^{−1}ν̃_{T} and showing they are equivalent .

First we compute ψ_{∗}^{−1}ν_{T}^{S^{1}} (h, ·) where h is any loop in L (S^{1}). Let t → X_{t·} is an L (ℝ)-valued Brownian motion. Let g satisfy the stochastic differential equation

g_{δts} = [(L_{g_{ts}})_{∗} ∂_{θ}] · X_{δts} with g_{0s} = 1,

as in Theorem 2.25. Here, since θ → (cos 2πθ, sin 2πθ) is our local chart,

(∂_{θ}F) (cos 2πθ, sin 2πθ) ≡ ∂_{θ}F (cos 2πθ, sin 2πθ) .

Then ν_{t}^{S^{1}} (e, ·) = Law h_{·}g_{t·} and thus

ψ_{∗}^{−1}ν_{T}^{S^{1}} (e, ·) = Lawψ^{−1} (h_{·}g_{T ·}) .

We claim g_{T ·} = ψ (X_{T ·}) and hence

ψ_{∗}^{−1}ν_{T}^{S^{1}} (h, ·) = Law [ψ^{−1} (h) (·) + X_{T ·}] .

To verify the claim that g_{t·} = ψ (X_{t·}), it will suffice to check that for any F ∈ C^{∞} (S^{1}, ℝ) we have

δ_{t}F (ψ (X_{ts})) = (∂_{θ}F) (ψ (X_{ts})) X_{δts}.

But by Ito’s Lemma δ_{t}F (ψ (X_{ts})) = (F ∘ ψ)′ (X_{δts}) X_{δts} = (∂_{θ}F) (ψ (X_{ts})) X_{δts} we are done. Thus

ψ_{∗}^{−1}ν_{T}^{S^{1}} (h, ·) = Law [ψ^{−1} (h) (·) + X_{T ·}] .

Since for fixed t, s → X_{ts} is a Brownian bridge in ℝ from 0 to 0 with parameter t, we have

(ψ_{∗}^{−1}ν_{T}^{S^{1}} (h, ·)) f = ∫ f (x + ψ^{−1} (h)) μ_{0,T}^{ℝ} (dx) .

Now we compute ψ_{∗}^{−1}μ_{0,T}^{S^{1}} explicitly. By Lemma 3.29 this is just μ_{ℤ,T}^{ℝ} or Wiener Measure conditioned on the integers (see Definition 3.26 and remark 3.27).

ON THE EQUIVALENCE OF MEASURES ON LOOP SPACE 31

Let f (x_{·}) ≡ F (x_{s_{1}}, ⋯, x_{s_{n}}) where F ∈ C^{∞} (ℝ^{n}) and 0 = s_{0} < ⋯ < s_{n} < 1. Then we have

(ψ_{∗}^{−1}μ_{0,T}^{S^{1}} ) [f]

= μ_{ℤ,T}^{ℝ} [f]

≡ ∫_{ℝ^{n}} \frac{F (x_{1}, ⋯, x_{n})}{P_{T}^{S^{1}} (e)} ∑_{α∈ℤ} P_{T (1−s_{n})}^{ℝ} (x_{n} + α) ∏_{i=1}^{n} P_{T △_{i}s}^{ℝ} (△_{i}x) dx_{i}

= ∑_{α∈ℤ} ∫_{ℝ^{n}} F (x_{1}, ⋯, x_{n}) \frac{P_{T (1−s_{n})}^{ℝ} (x_{n} + α)}{P_{T}^{S^{1}} (e)} ∏_{i=1}^{n} P_{T △_{i}s}^{ℝ} (△_{i}x) dx.

Also

(ψ_{∗}^{−1}ν̃_{T} ) [f]

= ∑_{α∈ℤ} C_{α} (ψ_{∗}^{−1}ν_{T}^{S^{1}} (σ_{α}, ·)) [f]

= ∑_{α∈ℤ} C_{α} ∫ f (x + ψ^{−1} (σ_{α})) μ_{0,T}^{ℝ} (dx)

= ∑_{α∈ℤ} ∫_{ℝ^{n}} F (x_{1} + αs_{1}, ⋯, x_{n} + αs_{n})

× \frac{C_{α}P_{T (1−s_{n})}^{ℝ} (x_{n})}{P_{T}^{ℝ} (0)} ∏_{i=1}^{n} P_{T △_{i}s}^{ℝ} (△_{i}x) dx_{i}

= ∑_{α∈ℤ} ∫_{ℝ^{n}} F (x_{1}, ⋯, x_{n}) \frac{C_{α}P_{T (1−s_{n})}^{ℝ} (x_{n} − αs_{n})}{P_{T}^{ℝ} (0)}

× ∏_{i=1}^{n} P_{T △_{i}s}^{ℝ} (△_{i}x − α△_{i}s) dx

= ∑_{α∈ℤ} ∫_{ℝ^{n}} F (x_{1}, ⋯, x_{n}) ∏_{i=1}^{n} P_{T △_{i}s}^{ℝ} (△_{i}x − α△_{i}s) dx

× \frac{1}{P_{T}^{S^{1}} (e)} exp (−\frac{α^{2}}{2T} ) P_{T (1−s_{n})}^{ℝ} (x_{n} − αs_{n}) .

Using Lemma 3.30 this last expression is just

= ∑_{α∈ℤ} ∫_{ℝ^{n}} F (x_{1}, ⋯, x_{n}) \frac{P_{1−s_{n}}^{ℝ} (x_{n} − α) ∏_{i=1}^{n} P_{△_{i}s}^{ℝ} (△_{i}x)}{P_{1}^{S^{1}} (e)}dx
= ∑_{α∈ℤ} ∫_{ℝ^{n}} F (x_{1}, ⋯, x_{n}) \frac{P_{1−s_{n}}^{ℝ} (x_{n} + α) ∏_{i=1}^{n} P_{△_{i}s}^{ℝ} (△_{i}x)}{P_{1}^{S^{1}} (e)}dx
= (ψ_{∗}^{−1}μ_{0}^{S^{1}}) f.

4. The Airault-Malliavin Theorem

In the next section we shall use the Airault-Malliavin Theorem (Theorem 4.1). For the reader’s convenience, we give a direct (and to our mind simpler) proof of this Theorem.

Let μ_{t} denote Wiener Measure on W_{e} (K) with parameter t and let μ_{0,t} be Pinned Wiener Measure as in Definition 2.9.

Theorem 4.1 (Airault & Malliavin, [26]). Recall from Definitions 2.9 and 2.7 that μ_{t} denotes Wiener measure on K with variance t and μ_{0,t} denotes pinned Wiener measure. Let △_{L(K)} be the operator from Definition 2.4 and let V_{t} : L (K) → ℝ denote the function

V_{t} (γ) = \frac{1}{2t^{2}} |∫_{0}^{1} γ (s)^{−1} γ (δs)|_{𝔎}^{2} − [\frac{dim 𝔎}{2t} + ∂_{t} log P_{t}^{K} (e)] ,

where the expression

∫_{0}^{1} γ (s)^{−1} γ (δs) ≡ lim_{ε→0} ∫_{0}^{1−ε} γ (s)^{−1} γ (δs) in L^{2} [μ_{0,t}] .

See Lemma 4.8 and Remark 4.9 Gross [17] for the existence of such a limit. Then for any smooth cylindrical function f : L (K) → ℝ (see Definition 2.5)

∂_{t}μ_{0,t} [f] = μ_{0,t} [\frac{1}{2}△_{L(K)} + V_{t}f] . (4.1)

We defer the proof of Theorem 4.1 until we have developed sufficient machinery. We shall be using some results of Gross. Accordingly we will need to define a few terms so that we can state some results from [17], [19].

Definition 4.2 (Notations from [17], [19]). The following definitions hold for Lemmas 4.4, 4.3 and 4.5:-

1. (Ω̂, P̂) be an abstract probability space and let t > 0.

2. Let Ê denote the expectation with respect to the measure P̂.

3. Let s → G_{s} be an arbitrary K-valued Brownian motion with parameter t starting from e (i.e. Law G = μ_{t}).

4. Define a 𝔎-valued Brownian motion β̃ by setting β̃_{α} ≡ ∫_{0}^{α} G (δs) G (s)^{−1}.

5. An element k in C ([0, 1] → K) is a finite energy path if

k′ (s) exists ds-a.s. and ∫_{0}^{1} |k^{−1}k′|_{𝔎}^{2} ds < ∞.

6. For any finite-energy path k define a μ_{t}-a.s. random variable J̃_{k} on W_{e} (K) by setting

J̃_{k} ∘ G = exp (−\frac{1}{2t} ∫_{0}^{1} |k^{−1}k′|_{𝔎}^{2} ds − \frac{1}{t} ∫_{0}^{1} ⟨k^{−1}k′, β̃_{δs}⟩_{𝔎}) .

Lemma 4.3 (Albeverio&Hoegh-Krohn, [3]). Let t = 1. Let k be a finite-energy path on K. Then for any bounded measurable f : W_{e} (K) → ℝ we have

Ê [f (G)] = Ê [f (kG) (J̃_{k} ∘ G)] .

This result goes through without trouble for any t > 0 (see Remark 4.6).

Lemma 4.4 (Gross:[19], Corollary 3.7). Let t = 1. Then for any finite-energy path k

∂_{ε} (J̃_{exp εh} ∘ G) |_{ε=0} = ∫_{0}^{1} \frac{1}{t} ⟨h′ (s) , β̃_{δs}⟩ ,

and the limit exists in L^{p} \binom{}{Ω} for any p < ∞. Let j̃_{h} denote the μ_{t}-a.s. random

variable so that j̃_{h} ∘ G = ∂_{ε} (J̃_{exp εh} ∘ G) |_{ε=0}. This result goes through without trouble for any t > 0 (see Remark 4.6).

Lemma 4.5 (Gross:[17], Lemma 4.8 and Remark 4.9). Let t = 1. Then for any p < ∞; β̃_{α} converges in L^{p} (μ_{t,0}) as α ↑ 1. Let β̃_{1} denote this limit in L^{2} (μ_{t,0}). [By Remark 4.6, this result goes through without trouble for any t > 0.

Remark 4.6 (Lemmas 4.4, 4.3 and 4.5 go though for any t > 0). Take t ≠ 1. Define a new Ad-invariant metric ˜⟨·, ·⟩ = \frac{1}{t} ⟨·, ·⟩_{𝔎} on 𝔎. Let \left\{{ \atop A}\right\} be a ˜⟨·, ·⟩-orthonormal

basis for 𝔎. Then 1 = ; ⟨Ã, Ã⟩ = \frac{1}{t} ⟨Ã, Ã⟩_{𝔎}. So \{Ã/\sqrt{t}\} is a ⟨·, ·⟩_{𝔎}-orthonormal

basis for 𝔎. Thus ˜△_{K}, the Laplacian on K̃ ≡ (K, ˜⟨·, ·⟩)is given by

˜△_{K} = ∑ ; Ã ∂2 ; Ã = t ∑ ; Ã ∂_{(Ã/\sqrt{t})}^{2} = t△_{K}.

So let G̃ be a standard Brownian motion on K̃. So let μ̃_{1} be Wiener Measure on K̃ with parameter 1 (i.e. μ̃_{1} = Law G̃). Then by the martingale characterization of a standard Brownian motion we have

df \binom{}{G_{s}} = dMartingale + \frac{1}{2}t (△_{K}f) \binom{}{G_{s}} ds.

In other words on K = (K, ⟨·, ·⟩_{𝔎}), G̃ is a Brownian motion with parameter t. Thus μ̃_{1} = μ_{t} and μ̃_{1,0} = μ_{t,0}. So applying Lemmas 4.5, 4.4 and 4.3 to K̃ we see that they extend to all t > 0.

Remark 4.7 (Our special case). For our purposes the space Ω̂ is W_{e} (K), the measure P̂ is Wiener measure μ_{t}, and the Brownian motion G_{s} is the map π_{s}^{−1}. Here π_{s} : W_{e} (K) → K is the map sending a path γ ∈ W_{e} (K) to an element γ_{s} ∈ K. We let G_{s} be π_{s}^{−1} rather than π_{s} because we shall need d (R_{g})_{∗} μ_{t}/dμ_{t} explicitly to compute derivatives whereas the theorems of Gross we cite use d (ℓ_{g})_{∗} μ_{t}/dμ_{t}. Recall that (ℓ_{g}γ) (s) = g (s) γ (s) , and (R_{g}γ) (s) = γ (s) g (s).

For the rest of this section, Ω̂, P̂, and G_{s} are to be interpreted as W_{e} (K) , μ_{t}, and π_{s}^{−1} respectively. For notational convenience, let π be the identity map from W_{e} (K) to itself and let π^{−1} denote the map from W_{e} (K) to itself taking a path γ to the path s → γ_{s}^{−1}.

Lemma 4.8 (The L^{2} (μ_{t})-adjoint ∂^{∗}). For any finite-energy path h ∈ H (𝔎) we have the L^{2} (μ_{t})-adjoint

∂_{h}^{∗} = −∂_{h} + ∫_{0}^{1} \frac{1}{t} ⟨h′ (s) , γ (s)^{−1} γ (δs)⟩ .

A more explicit statement is as follows:- Let f, g be elements of L^{∞−} (μ_{t}), where

L^{∞−} (μ_{t}) ≡ ∩_{p<∞}L^{p} (μ_{t}) .

Let R_{g} : W_{e} (K) → W_{e} (K) denote right multiplication by g (i.e. R_{g} (γ) (s) = γ (s) g (s)). Let L^{∞−} (μ_{t}) denote ∩_{p<∞}L^{p} (μ_{t}). Let D_{h} be the following domain:-

D_{h} ≡ \{u ∈ L^{∞−} (μ_{t}) : lim_{ε↓0} \frac{1}{ε} [u ∘ R_{exp εh} − u] exists in L^{p} (μ_{t}) , ∀p < ∞\} .

Let ∂_{h}u denote lim_{ε↓0} \frac{1}{ε} [u ∘ R_{exp εh} − u] for any u ∈ D_{h}. Define a μ_{t}-a.s. random variable j_{h} by setting

j_{h} (γ) = ∫_{0}^{1} \frac{1}{t} ⟨h′ (s) , γ (s)^{−1} γ (δs)⟩ .

Then

μ_{t} [g∂_{h}f] = −μ_{t} [f∂_{h}g] + μ_{t} [fgj_{h}] .

We will give a proof below. This result can also be obtained by using the left connection on K in Theorem 1.3 of Driver [10].

Proof. of Lemma 4.8

Let I : K → K given by I (κ) = κ^{−1} for any κ ∈ K. Abuse notation so that is g and γ are paths in W_{e} (K) then ℓ_{g} (γ) and R_{g} (γ) denote the paths s → g (s) γ (s) and s → γ (s) g (s) in W_{e} (K) respectively.

(4.2)  μ_{t} [g∂_{h}f] = μ_{t} [g lim_{ε↓0} \frac{1}{ε} (f ∘ R_{exp εh} − f)]
= lim_{ε↓0} \frac{1}{ε} (μ_{t} [gf ∘ R_{exp εh}] − μ_{t} [gf]) .

Apply Lemma 4.3 together with Remark 4.4 as well as the comments after Remark 4.7 to the finite-energy path k_{ε} = exp εh and the bounded measurable function f̃ = f ∘ I ∘ ℓ_{k^{−1}}. We obtain

Ê [f̃∘ π^{−1}] = Ê [f̃(kπ^{−1}) J̃_{k} ∘ π^{−1}] .

Upon simplification, we obtain

Ê [f ∘ R_{k} ∘ π] = Ê [(f ∘ π) (J̃_{k} ∘ π^{−1})] .

Here, as in Definition 4.2,

J̃_{k_{ε}} ∘ π^{−1} = exp (−\frac{1}{2t} ∫_{0}^{1} |k_{ε}^{−1}k′_{ε}|_{𝔎}^{2} ds − \frac{1}{t} ∫_{0}^{1} ⟨k_{ε}^{−1}k′_{ε}, π_{s}^{−1}π_{δs}⟩_{𝔎}) .

For any finite energy path k define a μ_{t}-a.s. random variable J_{k} by requiring that J_{k} ∘ π = J̃_{k} ∘ π^{−1}. Then we see that with respect to the measure μ_{t} we have that

J_{k_{ε}} (γ) = exp (−\frac{1}{2t} ∫_{0}^{1} |k_{ε}^{−1}k′_{ε}|_{𝔎}^{2} ds − \frac{1}{t} ∫_{0}^{1} ⟨k_{ε}^{−1}k′_{ε}, γ_{s}^{−1}γ_{δs}⟩_{𝔎}) ,

and

μ_{t} [f ∘ R_{k_{ε}}] = μ_{t} [fJ_{k_{ε}}] . (4.3)

Replacing f by gf ∘ R_{exp εh} and using Eq. [4.3] yields

μ_{t} [gf ∘ R_{exp εh}] = μ_{t} [(g ∘ R_{k_{ε}^{−1}}f) ∘ R_{k_{ε}}]
= μ_{t} [(g ∘ R_{k_{ε}^{−1}}) fJ_{k_{ε}}] .

Now we returning to Eq. [4.2] to get

μ_{t} [g∂_{h}f] = lim_{ε↓0} \frac{1}{ε} (μ_{t} [gf ∘ R_{exp εh}] − μ_{t} [gf])

lim_{ε↓0} ∫ f \frac{1}{ε} [(g ∘ R_{k_{ε}^{−1}}) J_{k_{ε}} − g] dμ_{t}.

Now by assumption g ∈ D_{h}, and so ∂_{ε}g ∘ R_{k_{ε}^{−1}} → ∂_{h}g in L^{p} (μ_{t}) , ∀p < ∞ as ε → 0.

By Lemma 4.4, Remark 4.6 we know that ∂_{ε} (J̃_{k_{ε}} ∘ π^{−1}) → (j̃_{h} ∘ π^{−1}) as ε → 0 in

L^{p} (μ_{t}) , ∀p < ∞. Notice that j_{h} ∘ π = j̃_{h} ∘ π^{−1} μ_{t}-a.s.. Thus ∂_{ε} (J_{k_{ε}} ∘ π) → (j_{h} ∘ π) in L^{p} (μ_{t}) , ∀p < ∞. Since Law π = μ_{t}, we have ∂_{ε}J_{k_{ε}} → j_{h} in L^{p} (μ_{t}) , ∀p < ∞ as ε → 0. Let o (ε) denote a family of functions so that \frac{1}{ε}o (ε) → 0 as ε → 0 in L^{p} (μ_{t}) , ∀p < ∞. Then

g ∘ R_{k_{ε}^{−1}} = g − ε∂_{h}g + o (ε) ,

and

J_{k_{ε}} = 1 + εj_{h} + o (ε) .

Therefore

\frac{1}{ε} [(g ∘ R_{k_{ε}^{−1}}) J_{k_{ε}} − g] f = −f∂_{h}g + fgj_{h} − εj_{h}f∂_{h}g + R (ε) ,

where the remainder R is given by

R (ε) = \frac{1}{ε}o (ε) (1 + g − ε∂_{h}g + o (ε) + εj_{h}) f.

Now j_{h}, f, ∂_{h}g are functions in L^{∞−} (μ_{t}). By using Hölder’s inequality repeatedly if necessary one can see that j_{h}f∂_{h}g and (1 + g − ε∂_{h}g + o (ε) + εj_{h}) f are also in L^{∞−} (μ_{t}). Hence μ_{t} [εj_{h}f∂_{h}g] → 0 and

[∫ R (ε) dμ_{t}]^{2} < μ_{t} [\frac{o (ε)}{ε}]^{2} μ_{t} [(1 + g − ε∂_{h}g + o (ε) + εj_{h}) f]^{2}
→ 0 as ε → 0.

Therefore

lim_{ε↓0} ∫ f \frac{1}{ε} [(g ∘ R_{k_{ε}^{−1}}) J_{k_{ε}} − g] dμ_{t} → μ_{t} [−f∂_{h}g + fgj_{h}] ,

and so we are done.

Corollary 4.9 (The L^{2} (μ_{t})-Adjoint (∂_{h}^{2})^{∗}). Let g, f be smooth cylinder functions (see Definition 2.5). Let h ∈ H (𝔎) be a finite energy path. Then

μ_{t}[g∂_{h}^{2}f] = μ_{t}[f∂_{h}^{2}g] − 2μ_{t}[j_{h}f∂_{h}g] + μ_{t}[fgj_{h}^{2}] − \frac{1}{t} |h|_{H(𝔎)}^{2} μ_{t}[gf]

Proof. By Lemma 4.8, we have

μ_{t}[g∂_{h}^{2}f] = −μ_{t}[∂_{h}g∂_{h}f] + μ_{t}[gj_{h}∂_{h}f]
(4.4)  = I_{1} + I_{2}.

Applying Lemma 4.8 to I_{1}, we see that

I_{1} = μ_{t}[f∂_{h}^{2}g] − μ_{t}[j_{h}f∂_{h}g]. (4.5)

To apply Lemma 4.8 to I_{2}, it will be necessary to show that gj_{h} is in the domain of the operator ∂_{h}; i.e. we must show [(gj_{h}) ∘ R_{exp εh} − gj_{h}] has a limit in L^{p} (μ_{t}) for any p < ∞. Since g is a smooth cylinder function, we already know that \frac{1}{ε} [g ∘ R_{exp εh} − g] converges to ∂_{h}g in L^{p} (μ_{t}). Thus, as in the proof of Lemma 4.8, if we can show that \frac{1}{ε} [j_{h} ∘ R_{exp εh} − j_{h}] converges to some ∂_{h}j_{h} in L^{p} (μ_{t}) for any p < ∞ then we will have

\frac{1}{ε} [(gj_{h}) ∘ R_{exp εh} − gj_{h}] → [g∂_{h}j_{h} + j_{h}∂_{h}g] (4.6)

as ε → 0 in L^{p} (μ_{t}) for any p < ∞.

From Lemma 4.8 recall that, for any finite-energy path h in H (𝔎), the random variable j_{h} is given by

j_{h} (γ) = ∫_{0}^{1} \frac{1}{t} ⟨h′ (s) , γ (s)^{−1} γ (δs)⟩ μ_{t}-a.s.

Let γ_{s} denote the μ_{t}-a.s. random variable γ (s). Thus j_{h} ∘ R_{exp εh} is given by

j_{h} ∘ R_{exp εh} (γ)

= ∫_{0}^{1} \frac{1}{t} ⟨h′ (s) , (γ_{s} exp εh (s))^{−1} δ_{s} (γ_{s} exp εh (s))⟩

= ∫_{0}^{1} \frac{1}{t} ⟨h′ (s) , exp (−εh (s)) γ_{s}^{−1}γ_{δs} exp εh (s)⟩

+ ∫_{0}^{1} \frac{1}{t} ⟨h′ (s) , exp (−εh (s)) exp′ [εh (s)] h′ (s) ds⟩ .

Therefore

\frac{1}{ε} [j_{h} ∘ R_{exp εh} − j_{h}] = ∫_{0}^{1} \frac{1}{tε} ⟨h′ (s) , Ad_{exp −εh_{s}} (γ_{s}^{−1}γ_{δs})⟩

− ∫_{0}^{1} \frac{1}{tε} ⟨h′ (s) , γ_{s}^{−1}γ_{δs}⟩

+ ∫_{0}^{1} \frac{1}{t} ⟨h′ (s) , exp (−εh_{s}) exp′ [εh (s)] h′ (s)⟩ ds.

By the Ad-invariance of the metric, we see that

\frac{1}{ε} [j_{h} ∘ R_{exp εh} − j_{h}] = ∫_{0}^{1} \frac{1}{tε} ⟨Ad_{exp εh_{s}}h′ (s) − h′ (s) , γ_{s}^{−1}γ_{δs}⟩

+ ∫_{0}^{1} \frac{1}{t} ⟨h′ (s) , exp (−εh_{s}) exp′ [εh (s)] h′ (s)⟩ ds.

However h was chosen to be good in the sense of Definition 2.2. So the matrix h (s) commutes with h′ (s) and thus Ad_{exp εh_{s}}h′ (s) − h′ (s) is 0. Observing this yields

\frac{1}{ε} [j_{h} ∘ R_{exp εh} − j_{h}] = ∫_{0}^{1} \frac{1}{t} ⟨h′ (s) , exp (−εh_{s}) exp′ (εh_{s}) h′ (s)⟩ ds.

which is independent of the path γ (and thus a constant random variable). Hence the above expression converges in L^{p} (μ_{t}) for all p < ∞ to the expression

∫_{0}^{1} \frac{1}{t} ⟨h′ (s) , exp (0) exp′ (0) h′ (s)⟩ ds = \frac{1}{t} |h|_{H(𝔎)}^{2} .

Thus ∂_{h}j_{h} = \frac{1}{t} |h′|_{H(𝔎)}^{2}. Returning to Eq. [4.6] we see that ∂_{h} (j_{h}g) exists and equals [g∂_{h}j_{h} + j_{h}∂_{h}g]. Thus

I_{2} = −μ_{t}[j_{h}f∂_{h}g] − \frac{1}{t} |h|_{H(𝔎)}^{2} μ_{t}[gf] + μ_{t}[fgj_{h}^{2}].

Now returning to Eqs. [4.4] and [4.5] we see that

μ_{t}[g∂_{h}^{2}f] = I_{1} + I_{2}
= μ_{t}[f∂_{h}^{2}g] − 2μ_{t}[j_{h}f∂_{h}g] − \frac{1}{t} |h|_{H(𝔎)}^{2} μ_{t}[gf] + μ_{t}[fgj_{h}^{2}].

Definition 4.10 (Orthogonal Decomposition of H (𝔎) and H_{0} (𝔎)). We will need the following notions:-

1. Recall from Definition 2.1 that

H (𝔎) ≡ {h : [0, 1] → 𝔎|h (0) = 0 and (h, h) < ∞} .

For any unit vector A ∈ 𝔎 and α in (0, 1) let Ã be the unit vector in H (𝔎) defined by setting

Ã(s) = \frac{1}{\sqrt{a}}A (s ∧ α) . (4.7)

Write H (𝔎) as U_{α}^{1} ⨁ U_{α}^{2} ⨁ U_{α}^{3} where the U_{α}^{i} are defined by setting

U_{α}^{1} ≡ {h ∈ H (𝔎) |h = 0 on [α, 1]};
U_{α}^{2} ≡ {h ∈ H (𝔎) |h = 0 on [0, α]};
U_{α}^{3} ≡ span ⟨Ã: A ∈ 𝔎⟩ .

Let S^{i} be a good orthonormal basis of U_{α}^{i}. Then S ≡ ∪_{i}S^{i} forms a good orthonormal basis of H (𝔎). Let △_{U_{α}^{i}} be defined as ∑_{h∈S^{i}} ∂_{h}^{2} where the operator (∂_{h}f) (γ) ≡ \frac{d}{dε}f (γ exp εh) [The map h → ∂_{h} is just the usual identification of elements of H (𝔎) with left-invariant vector fields on W_{e} (K)]. Then we can see that

△_{W_{e}(K)} = △_{U_{α}^{1}} + △_{U_{α}^{2}} + △_{U_{α}^{3}}. (4.8)

2. Recall from Definition 2.1 that

H_{0} (𝔎) ≡ {h ∈ H (𝔎) |h (1) = 0} .

Decompose H_{0} (𝔎) as W_{α}^{1} ⨁ W_{α}^{2} ⨁ W_{α}^{3}. W_{α}^{1} ≡ U_{α}^{1} which is defined as before. W_{α}^{2} is defined to be U_{α}^{2} ∩ H_{0} (𝔎). W_{α}^{3} is defined to be the span of the vectors ⟨ℓ_{A} : A ∈ 𝔎⟩ where the unit vector ℓ_{A} is given by setting

ℓ_{A} (s) = As1_{[0,α]}\sqrt{\frac{1 − α}{α}} + A (1 − s) 1_{(α,1]}\sqrt{\frac{α}{1 − α}}. (4.9)

. Let S_{0}^{i} be a good orthonormal basis of W_{α}^{i}. Then S_{0} ≡ ∪_{i}S_{0}^{i} forms a good orthonormal basis of H_{0} (𝔎). Let △_{W_{α}^{i}} be defined as ∑_{h∈S_{0}^{i}} ∂_{h}^{2} where the operator (∂_{h}f) (γ) ≡ \frac{d}{dε}f (γ exp εh) [The map h → ∂_{h} is just the usual identification of elements of H_{0} (𝔎) with left-invariant vector fields on L (K)]. Then we see that

△_{L(K)} = △_{U_{α}^{1}} + △_{W_{α}^{2}} + △_{W_{α}^{3}}. (4.10)

Proof. of Theorem 4.1:

Fix α < 1. Let f (σ) = F (σ_{s_{1}}, ⋯, σ_{s_{n}}) so thatσ_{s_{n}} < α. Let S = S^{1} ∪ S^{2} ∪ S^{3} be the orthonormal basis of Definition 4.10. Then

∂_{t}μ_{0,t} [f] = lim_{α→1} ∂_{t}μ_{t} [f \frac{P_{t(1−α)}^{K} ∘ π_{α}}{P_{t}^{K} (e)} ]

= lim_{α→1} μ_{t} [\frac{1}{2P_{t}^{K} (e)}△_{W_{e}(K)} (fP_{t(1−α)}^{K} ∘ π_{α})]

+ lim_{α→1} μ_{t} [f∂_{t} \frac{P_{t(1−α)}^{K} ∘ π_{α}}{P_{t}^{K} (e)} ]

(4.11)  = I + J.

Let us work on the second term first.

J = lim_{α→1} \frac{1}{P_{t}^{K} (e)}μ_{t} [f∂_{t}P_{t(1−α)}^{K} ∘ π_{α}] − μ_{0,t} [f∂_{t} log P_{t}^{K} (e)]
(4.12)  = J_{1} − μ_{0,t} [f∂_{t} log P_{t}^{K} (e)] ,

where

J_{1} = lim_{α→1} \frac{1}{P_{t}^{K} (e)}μ_{t} [f∂_{t}P_{t(1−α)}^{K} ∘ π_{α}]
(4.13)  = lim_{α→1} \frac{1 − α}{2P_{t}^{K} (e)}μ_{t} [f△_{K}P_{t(1−α)}^{K} ∘ π_{α}] .

Define

C_{α} (x_{n+1})

= ∫_{K^{n}} F (x_{1}, ⋯, x_{n}) P_{t(α−s_{n})}^{K} (x_{n}^{−1}x_{n+1}) ∏_{i=1}^{n} P_{t△_{i}s}^{K} (x_{i−1}^{−1}x_{i}) λ (dx_{i}) .

Then

lim_{α→1} μ_{t} [f△_{K}P_{t(1−α)}^{K} ∘ π_{α}] = lim_{α→1} ∫_{K} C_{α} (x) △_{K}P_{t(1−α)}^{K} (x) λ (dx)
= lim_{α→1} ∫_{K} △_{K}C_{α} (x) P_{t(1−α)}^{K} (x) λ (dx)
= △_{K}C_{1} (e)
< ∞.

From Eq. [4.13] we have

J_{1} = 0.

Combining this fact with Eq. [4.12] gives

J = −μ_{0,t} [f∂_{t} log P_{t}^{K} (e)] . (4.14)

We proceed to work on the first term I of Eq. [4.11].

I = lim_{α→1} \frac{1}{2}P_{t}^{K} (e)^{−1} μ_{t} [△_{W_{e}(K)} (fP_{t(1−α)}^{K} ∘ π_{α})]
= lim_{α→1} \frac{1}{2}μ_{0,t} [△_{W_{e}(K)}f] + lim_{α→1} \frac{1}{2}P_{t}^{K} (e)^{−1} μ_{t} [f△_{W_{e}(K)}P_{t(1−α)}^{K} ∘ π_{α}]

+ lim_{α→1} P_{t}^{K} (e)^{−1} ∑_{h∈S} μ_{t} [∂_{h}f∂_{h}P_{t(1−α)}^{K} ∘ π_{α}]

(4.15) = I_{1} + I_{2} + I_{3}.

From Eq. [4.8]

I_{1} = lim_{α→1} \frac{1}{2}μ_{0,t} [(△_{U_{α}^{1}} + △_{U_{α}^{2}} + △_{U_{α}^{3}}) f] .

Since f does not depend on the path on or after time α, △_{U_{α}^{2}}f = 0 and this last expression is

lim_{α→1} \frac{1}{2}μ_{0,t} [△_{U_{α}^{1}}f] + lim_{α→1} \frac{1}{2}μ_{0,t} [△_{U_{α}^{3}}f] .

Applying Eq. [4.10] and observing that △_{W_{α}^{2}}f = 0 reduces this last to

I_{1} = \frac{1}{2}μ_{0,t} [△_{L(K)}f] + lim_{α→1} \frac{1}{2}μ_{0,t} [△_{U_{α}^{3}}f] − lim_{α→1} \frac{1}{2}μ_{0,t} [△_{W_{α}^{3}}f] . (4.16)

Now letting A run through an orthonormal basis of 𝔎, we see from Definition 4.10 that

△_{W_{α}^{3}}f = ∑_{A} ∂_{ℓ_{A}}^{2}f.

Since f does not depend on the path from time α onwards, we can see that from Eqs. [4.7] and [4.9] that

∂_{ℓ_{A}}f = \sqrt{1 − α}∂Ãf and △_{W_{α}^{3}}f = (1 − α) △_{U_{α}^{3}}f.

Thus Eq. [4.16] becomes

I_{1} − \frac{1}{2}μ_{0,t} [△_{L(K)}f] = lim_{α→1} \frac{α}{2}μ_{0,t} [△_{U_{α}^{3}}f]
= lim_{α→1} \frac{1}{2P_{t}^{K} (e)}μ_{t} [P_{t(1−α)}^{K} ∘ π_{α} ∑_{A} ∂2 ; Ãf] .

Define β_{α} = ∫_{0}^{α} γ (s)^{−1} γ (δs), μ_{t}-a.s. Invoking Corollary 4.9 we obtain

μ_{t}[P_{t(1−α)}^{K} ∘ π_{α}∂2 ; Ãf] = μ_{t}[f∂2 ; ÃP_{t(1−α)}^{K} ∘ π_{α}] − 2μ_{t}[jÃf∂ÃP_{t(1−α)}^{K} ∘ π_{α}]

+μ_{t}[fP_{t(1−α)}^{K} ∘ π_{α}j2 ; Ã] − \frac{1}{t} |Ã|_{H(𝔎)}^{2} μ_{t}[P_{t(1−α)}^{K} ∘ π_{α}f].

Observing that Ã(s) = α^{−1/2} (s ∧ α) A, we see that |Ã|_{H(𝔎)}^{2} = 1, and jÃ (γ) =

\frac{1}{t}α^{−1/2} ⟨A, β_{α}⟩. Thus

I_{1} − \frac{1}{2}μ_{0,t} [△_{L(K)}f] = I_{2} + lim_{α→1} \frac{1}{2t^{2}α}μ_{t,e}[f |β_{α}|_{𝔎}^{2}] − \frac{dim 𝔎}{2t}μ_{t,e}[f]

− lim_{α→1} \frac{1}{tP_{t}^{K} (e)} ∑_{A} μ_{t}[⟨A, β_{α}⟩ f∂_{A}P_{t(1−α)}^{K} ∘ π_{α}].

Invoking Lemma 4.8 on I_{3} and recognizing that ∂_{h}P_{t(1−α)}^{K} ∘ π_{α} = 0 for any h ∈ S^{1} ∪ S^{2} yields

I_{3} = lim_{α→1} P_{t}^{K} (e)^{−1} ∑_{h∈S} μ_{t} [∂_{h}f∂_{h}P_{t(1−α)}^{K} ∘ π_{α}]
= − lim_{α→1} αP_{t}^{K} (e)^{−1} ∑_{h∈S} μ_{t} [f (△_{K}P_{t(1−α)}^{K}) ∘ π_{α}]

+ lim_{α→1} \frac{1}{tP_{t}^{K} (e)} ∑_{A} μ_{t}[⟨A, β_{α}⟩ f∂_{A}P_{t(1−α)}^{K} ∘ π_{α}]

= −2I_{2} + lim_{α→1} \frac{1}{tP_{t}^{K} (e)} ∑_{A} μ_{t}[⟨A, β_{α}⟩ f∂_{A}P_{t(1−α)}^{K} ∘ π_{α}].

Thus,

I = \frac{1}{2}μ_{0,t} [△_{L(K)}f] − \frac{dim 𝔎}{2t}μ_{t,e}[f] + lim_{α→1} \frac{1}{2t^{2}α}μ_{t,e}[f |β_{α}|_{𝔎}^{2}].

The expression

β_{α} = β_{α} ∘ π = β̃_{α} = ∫_{0}^{α} π_{s}^{−1}π_{δs}.

Combining Lemma 4.5 with Remarks 4.6 and 4.7 we have β̃_{α} converges in L^{2} (μ_{t,0}) as α ↑ 1. Thus β_{α} converges in L^{2} (μ_{t,0}) as α ↑ 1 to a limit β_{1} and so

I = \frac{1}{2}μ_{0,t} [△_{L(K)}f] − \frac{dim 𝔎}{2t}μ_{t,e}[f] + \frac{1}{2t^{2}} μ_{t,e}[f |β_{1}|_{𝔎}^{2}].

and so returning to Eqs. [4.11] and [4.12] yields

∂_{t}μ_{0,t} [f] = \frac{1}{2}μ_{0,t} [△_{L(K)}f] − \frac{dim 𝔎}{2t}μ_{t,e}[f] + \frac{1}{2t^{2}} μ_{t,e}[f |β_{1}|_{𝔎}^{2}] − μ_{0,t} [f∂_{t} log P_{t}^{K} (e)] .

5. Absolute continuity of Heat Kernel with respect to pinned Wiener measure

Let γ be a generic loop in L (K). Recall from Definitions 2.9 and 2.7 that μ_{t} (dγ) denotes Wiener measure on K with variance t and μ_{0,t} (dγ) denotes pinned Wiener measure. Recall from Definition 2.20 that ν_{t} (e, dγ) denotes Heat Kernel measure on L (K). The goal of this section is to demonstrate the absolute continuity of Heat Kernel measure ν_{t} (e, dγ) with respect to pinned Wiener measure μ_{0,t} (dγ).

Theorem 5.1. Heat Kernel measure ν_{t} (e, ·) on L (K) is absolutely continuous with respect to pinned Wiener measure μ_{0,t} (dγ) and the Radon-Nikodym derivative dν_{t} (e, ·) /dμ_{0,t} is bounded.

We defer the proof until some basic machinery is established.

Definition 5.2 (Basic Machinery). Let ℙ be the partition {0 < s_{1} < ⋯ < s_{n} < 1}. Then:-

1. g^{ℙ} ≡ (G_{0} (s_{i}, s_{j}))^{−1} as n × n matrices where G_{0} (s, σ) ≡ s ∧ σ − sσ was introduced in Definition 2.12.

2. Let A^{(i)} (x_{1}, ⋯, x_{n}) be the vector field on K^{n} so that

(A^{(i)}f) (x_{1}, ⋯, x_{n}) ≡ \frac{d}{dt}f (x_{1}, ⋯, x_{i} exp tA, ⋯, x_{n}) ↓_{t=0} .

3. ⟨·, ·⟩_{ℙ} is the left invariant metric on K^{n} such that ⟨A^{(i)}, B^{(j)}⟩_{ℙ} = δ_{AB}g_{ij}.

4. Let K^{ℙ} be K^{n} equipped with ⟨·, ·⟩_{ℙ} .

5. Let △_{ℙ} be the Laplacian on K^{ℙ}. Thus

△_{ℙ} = ∑_{A,i,j} G_{0} (s_{i}, s_{j}) ∂_{A^{(i)}}∂_{A^{(j)}}.

6. Let π_{s} : L (K) → K denote the map π_{s} : x → x_{s}.

7. Let π_{ℙ} : L (K) → K^{ℙ} by π_{ℙ} = (π_{s_{1}}, ⋯, π_{s_{n}}).

8. p_{t}^{ℙ} ≡ d (π_{ℙ})_{∗} ν_{t} (e, ·) /dλ. Explicitly if |ℙ| = n and λ denotes standard Haarmeasure on K, then p_{t}^{ℙ} : K^{n} → ℝ is the function so that for any F ∈ C^{∞} (K^{n}) we have

∫_{L(K)} F ∘ π_{ℙ} (γ) ν_{t} (e, dγ) = ∫_{x∈K^{n}} F (x) p_{t}^{ℙ} (x) λ^{⊗n} (dx) . (5.1)

9. q_{t}^{ℙ} ≡ d (π_{ℙ})_{∗} μ_{0,t}/dλ. Explicitly if |ℙ| = n and λ denotes standard Haarmeasure on K, then q_{t}^{ℙ} : K^{n} → ℝ is the function so that for any F ∈ C^{∞} (K^{n}) we have

∫_{L(K)} F ∘ π_{ℙ} (γ) μ_{0,t} (dγ) = ∫_{x∈K^{n}} F (x) q_{t}^{ℙ} (x) λ^{⊗n} (dx) . (5.2)

10. 𝔉^{ℙ} ≡ σ ⟨π_{ℙ}⟩ .

11. Z_{t}^{ℙ} ≡ (p_{t}^{ℙ}/q_{t}^{ℙ}) ∘ π_{ℙ} = d (ν_{t}|_{𝔉^{ℙ}}) /d (μ_{0,t}|_{𝔉^{ℙ}}). Explicitly for any F ∈ C^{∞} (K^{n}), combining Eqs. [5.1] and [5.2] we have

∫ F ∘ π_{ℙ} (γ) ν_{t} (e, dγ) = ∫ F \frac{p_{t}^{ℙ}}{p_{t}^{ℙ}} (q_{t}^{ℙ}dλ^{⊗n}) = ∫ F ∘ π_{ℙ} (γ) Z_{t}^{ℙ} (γ) μ_{0,t} (dγ) .

The proof of Theorem 5.1 rests on Theorem 4.1, a result of Airault and Malliavin. We shall also make use of Lemma 2.15.

Lemma 5.3 (Asymptotic properties of heat Kernels on K). Heat Kernel measure on K has the following properties:-

1. ε^{d/2}P_{ε}^{K} (e) → (2π)^{−d/2} as ε → 0.

2. Let B^{K} (e, ε) denote the ball of radius ε near e. Then

sup_{ε∈(0,1),u∈B^{K}(e,ε)^{c}} P_{ε(1−s_{n})}^{K} (u) < ∞.

Proof. A generalization of this result is proved in Berline, Getzler, & Vergne, [6], Theorem 2.30. See also [28].

Lemma 5.4 (μ_{0,ε} → δ_{e} as ε → 0). Let f : K^{n} → ℝ be continuous. Abusing notation, let e denote the element (e, ⋯, e) of K^{n}. Then

lim_{ε→0} (π_{ℙ})_{∗} μ_{0,ε} [f] = f (e) .

Proof. Let π_{s} be the evaluation map as in Definition 5.2. Let △_{i}s be s_{i} − s_{i−1} as usual. Then

lim_{ε→0} (π_{ℙ})_{∗} μ_{0,ε} [f] = lim_{ε→0} ∫_{M} f (x_{1}, ⋯, x_{n}) q_{ε}^{ℙ} (x) dλ

= lim_{ε→0} ∫_{M} f (x_{1}, ⋯, x_{n}) \frac{P_{ε(1−s_{n})}^{K} (x_{n})}{P_{ε}^{K} (e)} ∏_{i=1}^{n} P_{ε△_{i}s}^{K} (x_{i−1}^{−1}x_{i}) dx_{i}

= lim_{ε→0} μ_{0,ε} [f ∘ π_{ℙ} \frac{P_{ε(1−s_{n})}^{K} ∘ π_{s_{n}}}{P_{ε}^{K} (e)} ] .

Let B^{K} (e, r) be the open ball of all points distant less than r from e in the metric ⟨·, ·⟩_{𝔎} on K. Then our previous expression becomes

= lim_{ε→0} μ_{ε} [\frac{f ∘ π_{ℙ}}{P_{ε}^{K} (e)} (1_{B^{K}(e,ε)} ∘ π_{s_{n}}) (P_{ε(1−s_{n})}^{K} ∘ π_{s_{n}})]

+ lim_{ε→0} μ_{ε} [\frac{f ∘ π_{ℙ}}{P_{ε}^{K} (e)} (1_{B^{K}(e,ε)^{c}} ∘ π_{s_{n}}) (P_{ε(1−s_{n})}^{K} ∘ π_{s_{n}})]

= I_{1} + I_{2}.

By Lemma 5.3, we see that the expression

\frac{f (x_{1}, ⋯, x_{n})}{P_{ε}^{K} (e)}1_{B^{K}(e,ε)^{c}} (x_{n}) P_{ε(1−s_{n})}^{K} (x_{n})

is bounded and so I_{2} vanishes by Dominated Convergence. Thus

lim_{ε→0} (π_{ℙ})_{∗} μ_{0,ε} [f]

= lim_{ε→0} μ_{ε} [\frac{f ∘ π_{ℙ}}{P_{ε}^{K} (e)} (1_{B^{K}(e,ε)} ∘ π_{s_{n}}) (P_{ε(1−s_{n})}^{K} ∘ π_{s_{n}})]

= lim_{ε→0} (π_{ℙ})_{∗} μ_{0,ε} [f (x_{1}, ⋯, x_{n−1}, e)]

+ lim_{ε→0} (π_{ℙ}^{∗}μ_{0,ε}) [(f (x_{1}, ⋯, x_{n}) − f (x_{1}, ⋯, x_{n−1}, e)) 1_{B^{K}(e,ε)} (x_{n})]

= J_{1} + J_{2}.

Now the expression

sup_{K^{ℙ}} |f (x_{1}, ⋯, x_{n}) − f (x_{1}, ⋯, x_{n−1}, e)| 1_{B^{K}(e,ε)} (x_{n})

is bounded above by

sup_{x_{n}∈B^{K}(e,ε)} |f (x_{1}, ⋯, x_{n}) − f (x_{1}, ⋯, x_{n−1}, e)| . (5.3)

We claim that if x_{n} is distance d from e in K then there is a constant C^{ℙ}, depending only on the partition, so that the points (x_{1}, ⋯, x_{n}) and (x_{1}, ⋯, x_{n−1}, e) are distance dC^{ℙ} apart in K^{ℙ}. If we can verify this, then Eq.[5.3] is bounded above by

sup_{{x,y|y∈B^{ℙ}(x,ε/C^{ℙ})}} |f (y) − f (x)| .

By the uniform continuity of the continuous function f on the compact topological space K^{ℙ} we see that this last expression tends to 0 as ε → 0. Thus J_{2} can be made arbitrarily small. This, in turn, implies that

lim_{ε→0} (π_{ℙ})_{∗} μ_{0,ε} [f] = lim_{ε→0} (π_{ℙ})_{∗} μ_{0,ε} [f (x_{1}, ⋯, x_{n−1}, e)] .

Now replace ℙ by ℙ_{1} ≡ {0 < s_{1} ⋯ < s_{n−1} < 1} and f by

f_{1} (x_{1}, ⋯, x_{n−1}) ≡ f (x_{1}, ⋯, x_{n−1}, e) .

f_{1} is still smooth on K^{ℙ_{1}} and the above reasoning applies inductively. Therefore

lim_{ε→0} (π_{ℙ})_{∗} μ_{0,ε} [f] = f (e, ⋯, e) ,

and we are done once we verify the claim.

To do this, let x (s) ≡ (c_{1}, ⋯, c_{n−1}, x_{n} (s)) be a differentiable path in K^{ℙ} (with the c_{i} being held constant) then we have

x′ (s) = ∑_{A} ⟨x′_{n} (s) , A⟩_{𝔎} A^{(n)}

for any s. This implies that

∫_{0}^{1} |x′ (s)|_{ℙ}^{2} ds = ∫_{0}^{1} ∑_{A} ⟨x′_{n} (s) , A⟩_{𝔎}^{2} |A^{(n)} (s)|_{ℙ}^{2} ds

= (G_{0} (s_{i}, s_{j}))_{nn}^{−1} ∫_{0}^{1} |x′_{n} (s)|_{𝔎}^{2} ds
= (C^{ℙ})^{2} ∫_{0}^{1} |x′_{n} (s)|_{𝔎}^{2} ds.

Thus if x_{n} is distance d from e in K then the points (x_{1}, ⋯, x_{n}) and (x_{1}, ⋯, x_{n−1}, e) are distance dC^{ℙ} apart in K^{ℙ}.

Lemma 5.5. Let C_{t} ≡ log [t^{d/2}P_{t}^{K} (e)] − log lim_{ε→0} ε^{d/2}P_{ε}^{K} (e). C_{t} is well-defined by Lemma 5.3. Then p_{t}^{ℙ}/q_{t}^{ℙ} ≤ exp C_{t}.

Proof. Given bounded smooth f, h ≥ 0, on K^{n} define

H (t, x) ≡ ∫ h (xy) p_{t}^{ℙ} (y) dy
F (t, x) ≡ ∫ f (xy) q̃_{t}^{ℙ} (y) dy.

Let γ ∈ L (K) and let t → g_{t} be our standard Brownian motion on L (K) (see Definition 2.22). Let ℓ_{x} : k → xk denote left translation by x on K. Now t → γg_{t} is a Brownian motion on L (K) starting at γ in the sense of Definition 2.14. Heat Kernel measure, ν_{t} (γ, ·) is the law of γg_{t}.

ν_{t} (γ, h ∘ π_{ℙ}) = Eh ∘ π_{ℙ} (γg_{t})
= Eh (π_{ℙ} (γ) π_{ℙ} (g_{t}))
= Eh ∘ ℓ_{π_{ℙ}(γ)} ∘ π_{ℙ} (g_{t})
= ν_{t} (e, h ∘ ℓ_{π_{ℙ}(γ)} ∘ π_{ℙ})
= ∫ h (π_{ℙ} (γ) y) p_{t}^{ℙ} (y) dy
= H (t, π_{ℙ} (γ)) .

Letting x = π_{ℙ} (γ), we see from the Heat Equation, Theorem 2.21, and Lemma 2.15 that

∂_{t}H (t, x) = \frac{1}{2}△_{L(K)}H (t, π_{ℙ} (γ)) = \frac{1}{2}△_{ℙ}H (t, x)
(5.4)  H (t, x) → h ∘ π_{ℙ} (γ) as t → 0.

We shall now obtain a similar equation for F (t, x). Let q̃_{t}^{ℙ} ≡ q_{t}^{ℙ} exp C_{t}. Then for some smooth ϕ on K^{n}, we have

∫ ϕ (y) (∂_{t}q_{t}^{ℙ}) (y) dy = ∂_{t} exp (C_{t}) μ_{0,t} [ϕ ∘ π_{ℙ}]
= exp (C_{t}) ∂_{t}μ_{0,t} [ϕ ∘ π_{ℙ}]

+μ_{0,t} [ϕ ∘ π_{ℙ}] exp (C_{t}) [\frac{dim 𝔎}{2t} + ∂_{t} log p_{t}^{K} (e)] .

Applying Airault-Malliavin (Theorem 4.1) to the first term yields

= exp (C_{t}) μ_{0,t} [\frac{1}{2t^{2}} |∫_{0}^{1} x (ds) x (s)^{−1}|_{𝔎}^{2} ϕ ∘ π_{ℙ}]

+\frac{exp (C_{t})}{2}μ_{0,t} [△_{L(K)}ϕ ∘ π_{ℙ}] .

Define Γ_{t}^{ℙ} ∘ π_{ℙ} ≡ μ_{0,t} (|∫_{0}^{1} γ_{ds}γ_{s}^{−1}|_{𝔎}^{2} |𝔉^{ℙ}) where |∫_{0}^{1} γ_{ds}γ_{s}^{−1}|_{𝔎} is Gross’ L^{2} (μ_{0,t})-

limit of |∫_{0}^{α} γ_{ds}γ_{s}^{−1}|_{𝔎} as α → 1. Then using Lemma 2.15 our last expression becomes

= exp (C_{t}) (π_{ℙ})_{∗} μ_{0,t} [\frac{Γ_{t}^{ℙ}}{2t^{2}} ϕ + \frac{1}{2}△_{ℙ}ϕ]
= exp (C_{t}) ∫ [\frac{Γ_{t}^{ℙ} (y)}{2t^{2}}ϕ (y) + \frac{1}{2} (△_{ℙ}ϕ) (y)] q_{t}^{ℙ} (y) dy
= ∫ \frac{Γ_{t}^{ℙ} (y)}{2t^{2}}ϕ (y) q_{t}^{ℙ} (y) dy + \frac{1}{2} (△_{ℙ}ϕ) (y) q_{t}^{ℙ} (y) dy.

Using the smoothness of q_{t}^{ℙ}, perform an integration by parts on the second term to get

= ∫ \frac{Γ_{t}^{ℙ} (y)}{2t^{2}}ϕ (y) q_{t}^{ℙ} (y) dy + \frac{1}{2} (△_{ℙ}q_{t}^{ℙ}) (y) ϕ (y) dy
= ∫ [\frac{Γ_{t}^{ℙ} (y)}{2t^{2}}q_{t}^{ℙ} (y) + \frac{1}{2} (△_{ℙ}q_{t}^{ℙ}) (y)] ϕ (y) dy.

Therefore we have the dy-a.s. equality

(∂_{t}q_{t}^{ℙ}) (y) = \frac{Γ_{t}^{ℙ} (y)}{2t^{2}}q_{t}^{ℙ} (y) + \frac{1}{2} (△_{ℙ}q_{t}^{ℙ}) (y) . (5.5)

Let s → β_{s} be a standard Brownian motion on K with parameter t. Then for any continuous ϕ on K^{n} we have

∫ ϕ (y) q̃_{t}^{ℙ} (y) dy = exp C_{t}μ_{0,t} [ϕ ∘ π_{ℙ}]

= exp C_{t}μ_{t} [ϕ ∘ π_{ℙ} \frac{P_{t(1−s_{n})}^{K} ∘ π_{s_{n}}}{P_{t}^{K} (e)} ]

= exp C_{t}E [ϕ (β_{s_{1}} ⋯ β_{s_{n}}) \frac{P_{t(1−s_{n})}^{K} (β_{s_{n}})}{P_{t}^{K} (e)}] .

Notice that s → β_{s}^{−1} is also a standard Brownian motion on K. This means that our previous expression becomes

= exp C_{t}E [ϕ (β_{s_{1}}^{−1} ⋯ β_{s_{n}}^{−1}) \frac{P_{t(1−s_{n})}^{K} (β_{s_{n}}^{−1})}{P_{t}^{K} (e)}] .

Now using the fact that P_{t}^{K} (x) = P_{t}^{K} (x^{−1}) on K yields

= exp C_{t}E [ϕ (β_{s_{1}}^{−1} ⋯ β_{s_{n}}^{−1}) \frac{P_{t(1−s_{n})}^{K} (β_{s_{n}})}{P_{t}^{K} (e)}]

= exp C_{t} ∫ ϕ (γ_{s_{1}}^{−1} ⋯ γ_{s_{n}}^{−1}) μ_{0,t} (dγ)
= ∫ ϕ (y^{−1}) q̃_{t}^{ℙ} (y) dy
= ∫ ϕ (y) q̃_{t}^{ℙ} (y^{−1}) dy.

Thus for any continuous ϕ on K^{n} we have

∫ ϕ (y) q̃_{t}^{ℙ} (y) dy = ∫ ϕ (y) q̃_{t}^{ℙ} (y^{−1}) dy. (5.6)

Using this yields

F (t, x) = ∫ f (xy) q̃_{t}^{ℙ} (y) dy
= ∫ f (xy) q̃_{t}^{ℙ} (y^{−1}) dy
= ∫ f (y) q̃_{t}^{ℙ} (y^{−1}x) dy.

Applying Eq. [5.5] to compute the derivative ∂_{t}F (t, x) yields

∂_{t}F (t, x) = ∫ f (y) ∂_{t}q̃_{t}^{ℙ} (y^{−1}x) dy
= ∫ f (y) \frac{Γ_{t}^{ℙ} (y^{−1}x)}{2t^{2}}q_{t}^{ℙ} (y^{−1}x) dy + \frac{1}{2} ∫ f (y) (△_{ℙ}q_{t}^{ℙ}) (y^{−1}x) dy
= I (t, x) + \frac{1}{2} ∫ f (y) (△_{ℙ}q_{t}^{ℙ}) (y^{−1}x) dy.

46 VIKRAM K. SRIMURTHY

By the left-invariance of the Laplacian, △_{ℙ} this last expression is

= I (t, x) + \frac{1}{2} ∫ f (y) (△_{ℙ}q_{t}^{ℙ} ∘ ℓ_{y^{−1}}) (x) dy
= I (t, x) + \frac{△_{ℙ}}{2} ∫ f (y) q_{t}^{ℙ} (y^{−1}x) dy
= I (t, x) + \frac{△_{ℙ}}{2} ∫ f (xy) q_{t}^{ℙ} (y^{−1}) dy.

Using Eq. [5.6] a second time yields

= I (t, x) + \frac{△_{ℙ}}{2} ∫ f (xy) q_{t}^{ℙ} (y) dy
= I (t, x) + \frac{△_{ℙ}}{2} F (t, x) .

Therefore

∂_{t}F (t, x) = I (t, x) + \frac{△_{ℙ}}{2} F (t, x) .

As t gets small, we have

lim_{t→0} F (t, x) = lim_{t→0} ∫ f (xy) q̃_{t}^{ℙ} (y) dy
= lim_{t→0} exp C_{t} ∫ f (xy) q_{t}^{ℙ} (y) dy
= lim_{t→0} (π_{ℙ})_{∗} μ_{0,t} [f ∘ ℓ_{x}]
= f (x) by Lemma 5.4.

Thus we have

∂_{t}F (t, x) = I (t, x) + \frac{△_{ℙ}}{2} F (t, x)
(5.7)  F (t, x) → f (x) as t → 0.

We are now ready to apply the Duhammel’s principle. For any η > 0, pick f (x) = η + h (x) ≥ 0. Since f ≥ 0 a.s., I ≥ 0, since Γ_{t}^{ℙ} ∘ π_{ℙ} is the conditional expectation of the positive function |∫_{0}^{1} γ_{ds}γ_{s}^{−1}|_{𝔎}^{2}. Let

U (t, x) ≡ (F − H) (t, x) .

Using the fact that f (x) = η + h (x) implies that U (t, x) = η exp C_{t} + ∫ h (xy) (q̃_{t}^{ℙ} − p_{t}^{ℙ}) (y) dy. (5.8) By Eqs. [5.4] and [5.7] we see that

∂_{t}U (t, x) = \frac{1}{2}△_{ℙ}U (t, x) + I (t, x) . (5.9)

Formally guessing a solution by Duhammel’s principle let us define

Ũ (t, x) ≡ exp (\frac{t − ε}{2}△_{ℙ}) U (ε, x) + ∫_{ε}^{t} exp (\frac{t − τ}{2} △_{ℙ}) I (τ, x) dτ.

Then Ũ (ε, x) = U (ε, x) and

∂_{t}Ũ (t, x) = \frac{1}{2}△_{ℙ} exp (\frac{t − ε}{2}△_{ℙ}) U (ε, x) + I (t, x)

+ ∫_{ε}^{t} \frac{△_{ℙ}}{2} exp (\frac{t − τ}{2} △_{ℙ}) I (τ, x) dτ

= \frac{1}{2}△_{ℙ}Ũ (t, x) + I (t, x) .

Therefore U = Ũ and so

U (t, x) = exp (\frac{t − ε}{2}△_{ℙ}) U (ε, x) + ∫_{ε}^{t} exp (\frac{t − τ}{2} △_{ℙ}) I (τ, x) dτ. (5.10)

Let t_{0} be “the first time that U goes below zero”. Explicitly

t_{0} ≡ inf {t > 0| inf_{x} U (t, x) < 0} . (5.11)

If we can show that t_{0} > 0 then we can take ε in Eq.[5.10] equal to t_{0}/2. Then since U (ε, ·) and I are non-negative we must have U (t, x) ≥ 0 for any t and x. Then we can let η → 0 in Eq. [5.8] to obtain

∫ h (xy) (q̃_{t}^{ℙ} − p_{t}^{ℙ}) (y) dy ≥ 0.

Then q̃_{t}^{ℙ} − p_{t}^{ℙ} will be non-negative almost surely and so p_{t}^{ℙ} ≤ q_{t}^{ℙ} exp C_{t} and we shall be done.

Thus the problem reduces to showing t_{0} > 0. Suppose t_{0} = 0. There exist times τ_{i}, τ_{i} > 0, τ_{i} ↓ 0 as i → ∞ so that inf_{x} U (τ_{i}, x) < −1/i. By the compactness of K^{n}, for each τ_{i} there must exist an x_{i} so that

U (τ_{i}, x_{i}) = inf_{x} U (τ_{i}, x) (τ_{i}, x) < −1/i.

Thus by compactness, there exist a convergent subsequence of the x_{i} So without losing generality, suppose x_{i} → x_{∞} in K^{n}. Then (τ_{i}, x_{i}) → (0, x_{∞}) in [0, ∞) × K^{n} so that U (τ_{i}, x_{i}) → (f − h) (x_{∞}) = η > 0. But U (τ_{i}, x_{i}) ≤ 0 for all i which implies that (f − h) (x_{∞}) ≤ 0 giving us our contradiction. Thus t_{0} > 0 and we are done.

We are now able to return to the proof of Theorem 5.1.

Proof. of Theorem 5.1

Let {ℙ_{n}} be a refining sequence of partitions of (0, 1) (i.e. one is not allowed to include the endpoints 0, 1 in the partition) such that |ℙ_{n}| → 0. Since ℙ_{n} a refining sequence, Z_{t}^{ℙ_{n}} is a non-negative discrete 𝔉^{ℙ_{n}} martingale, where 𝔉^{ℙ_{n}} ≡ σ ⟨π_{ℙ_{n}}⟩.

To make this clear, let n > m and f ∈ 𝔉^{ℙ_{m}}. Then can find smooth functions F_{1} : K^{ℙ_{m}} → ℝ and F_{2} : K^{ℙ_{n}} → ℝ so that f = F_{1} ∘ π_{ℙ_{m}} = F_{2} ∘ π_{ℙ_{n}}. Now

μ_{0,t}Z_{t}^{ℙ_{n}}f = (π_{ℙ_{n}}^{∗}μ_{0,t}) F_{2}p_{t}^{ℙ_{n}}/q_{t}^{ℙ_{n}} = (π_{ℙ_{m}}^{∗}ν_{t}) F_{1} = μ_{0,t}Z_{t}^{ℙ_{m}}f,

which shows that Z_{t}^{ℙ_{n}} is a discrete 𝔉^{ℙ_{n}} martingale.

Suppose we can show

sup_{n} ∥Z_{t}^{ℙ_{n}} ∥_{L^{2}(μ_{0,t})} < ∞. (5.12)

Then we have {Z_{t}^{ℙ_{n}}} is uniformly integrable since

lim_{M→∞} sup_{n} μ_{0,t} [Z_{t}^{ℙ_{n}}1_{\{Z_{t}^{ℙn} >M\}}] ≤ lim_{M→∞} \frac{1}{M} sup_{n} μ_{0,t} (Z_{t}^{ℙ_{n}})^{2} = 0.

The fact that {Z_{t}^{ℙ_{n}}} a uniformly-integrable discrete L^{1}-martingale implies the following (see Durrett [14]):-

1. Z_{t}^{ℙ_{n}} converges in L^{1}.

2. If Z_{t} ≡ lim_{n→∞} Z_{t}^{ℙ_{n}} then μ_{0,t} (Z_{t}|𝔉^{ℙ_{n}}) = Z_{t}^{ℙ_{n}}.

But now μ_{0,t} [Z_{t}f ∘ π_{ℙ_{n}}] = ν_{t}f ∘ π_{ℙ_{n}} for any n ∈ N. Hence Z_{t} must be dν_{t}/dμ_{0,t} on the desired σ-algebra ⟨x_{t} : γ_{·} → γ_{t} : t ∈ [0, 1]⟩ .

So the problem reduces to proving Eq. (5.12).

To this end, pick an arbitrary partition ℙ ≡ {0 < s_{1} < ⋯ < s_{n} < 1} and let f be smooth on K^{ℙ}.We want to compute

∥Z_{t}^{ℙ}∥_{L^{2}(μ_{0,t})}^{2} = π_{ℙ_{n}}^{∗}μ_{0,t} (p_{t}^{ℙ}/q_{t}^{ℙ})^{2} .

But by Lemma 5.5, p_{t}^{ℙ}/q_{t}^{ℙ} ≤ exp C_{t} where C_{t} is finite and defined by Lemma 5.5. We have a stronger condition than Eq. (5.12). Hence we are done. Moreover Z_{t} < exp C_{t}.

6. Semi-Martingale Properties of g_{T,·}

Let Ω = C ([0, 1] → L (K)) be our probability space, let P be the law of a Brownian motion on L (K), and let g_{t} : Ω → L (K) be the evaluation map at t as in Definition 2.22. Then t → g_{t} is an L (K)-valued Brownian motion and thus Law g_{t} equals Heat Kernel measure ν_{t} (e, ·).

Remark 6.1 (g_{t·} is a semimartingale). In Section 5 we showed that Heat kernel measure ν_{t} (e, ·) is absolutely continuous with respect to pinned Wiener measure μ_{0,t}. Let γ_{s} : L (K) → K be the evaluation map at time s. Then equipping L (K) with pinned Wiener measure μ_{0,t}, we see that s → γ_{s} is a Brownian bridge and thus a semimartingale. Since ν_{t} (e, ·) ≪ μ_{0,t} and s → γ_{s} is a μ_{0,t}-semimartingale, we know (see Theorem 2, page 45 of [30]) that s → γ_{s} is a ν_{t} (e, ·)-semimartingale. Now the random variables (γ, ν_{t} (e, ·)) and (g_{t}, P) share the same law. Therefore, by Definition 2.24 we see that s → g_{ts} is an 𝔉_{t·}-semimartingale.

In this section we provide an explicit decomposition for the 𝔉_{ts}-semimartingale s ↦ g_{ts} and compute its pullback ∫_{0}^{s} g_{tδσ}g_{tσ}^{−1} . We do this by approximating g_{t·} by the piecewise C^{1} functions g_{t·}^{ℙ} (described in Definition 6.9). Then we compute the approximate pullback ∫_{0}^{·} (∂_{s}g_{ts}^{ℙ}) (g_{ts}^{ℙ})^{−1} ds (which is a semimartingale since it is piecewise C^{1}). Then as a result of Propositions 6.13, 6.19, 6.20, and 6.21 we show that these approximations converge to a process Y_{ts}. In Theorem 6.11 we then show that g satisfies g_{t·} = 1 + ∫_{0}^{·} Y_{tδs}g_{ts}.

We care about the pullback Y_{t·} because we will show in Section 7 that it has a law to equivalent to that of a Brownian motion on a restricted σ-algebra. This will then imply that Pinned Wiener measure is absolutely continuous with Heat Kernel measure on σ ⟨γ_{s}|s ∈ [0, 1 − ε)⟩ for any ε > 0 (Recall γ_{s} is evaluation at time s).

6.1. Preliminaries.

6.1.1. Two parameter processes. We will need to introduce two-parameter stochastic integration in order to state our main result, Theorem 6.11.

Remark 6.2 (The Brownian sheet b generates the filtration). From Definition 2.22 we see that 𝔉_{ts} = σ ⟨g_{τσ}|τ ≤ t or σ ≤ s⟩ ∨ 𝔉_{00}. From Theorem 2.25 we see that

g satisfies g_{δts} = g_{ts}X_{δts} with g_{0s} = e and X_{ts} = ∫ g_{δts}g_{ts}^{−1}.

Therefore g_{ts} is in the σ-algebra generated by the random variables

⟨X_{τσ} : (τ, σ) ∈ [0, t] × [0, s]⟩ ,

and X_{ts} is in the σ-algebra generated by the variables

⟨g_{τσ} : (τ, σ) ∈ [0, t] × [0, s]⟩ .

Again by Theorem 3.19, b_{ts} is in the σ-algebra generated by the

⟨X_{τσ} : (τ, σ) ∈ [0, t] × [0, s]⟩ ,

while X_{ts} is in the σ-algebra generated by

⟨b_{τσ} : (τ, σ) ∈ [0, t] × [0, s]⟩ .

Therefore 𝔉_{ts} = σ ⟨b_{τσ}|τ ≤ t or σ ≤ s⟩ ∨ 𝔉_{00}. This observation is important to use the results of Cairoli and Walsh in [7].

Definition 6.3 (Cairoli & Walsh [7]). We will use the following notions from Cairoli & Walsh:-

1. Let (Ω, {𝔉_{ts}}) be our probability space from Definition 2.22.

2. 𝔉_{ts}^{1} ≡ 𝔉_{t1} ∨ 𝔉_{1s}. This is the σ-algebra generated by ⟨b_{τσ}|τ ≤ t or σ ≤ s⟩ ∨ 𝔉_{00} from Remark 6.2.

3. Let b be the Brownian sheet from Theorem 3.19 and let b^{A} ≡ ⟨b, A⟩_{𝔎} for any A ∈ 𝔎.

4. For t_{1}s_{1} t_{1} < t_{2} and s_{1} < s_{2}, let (t_{1}s_{1}, t_{2}s_{2}] denote the rectangle (t_{1}, t_{2}] × (s_{1}, s_{2}].

5. R_{ts} ≡ (0, t] × (0, s] = (00, ts].

6. Let L be the set of all ℝ-valued processes ϕ : [0, 1]×[0, 1]×Ω, so that ϕ_{ts} (·) ≡ ϕ (t, s, ·) is 𝔉_{ts}-measurable and the expectation E ∫_{0}^{1} ∫_{0}^{1} ϕ_{ts}^{2} (g) dsdt < ∞.

7. For any ϕ ∈ L, define ϕ ((t_{1}s_{1}, t_{2}s_{2}]) ≡ ϕ_{t_{2}s_{2}} − ϕ_{t_{2}s_{1}} − ϕ_{t_{1}s_{2}} + ϕ_{t_{1}s_{1}}.

8. A two-parameter process M ∈ L such that M vanishes on the axes (i.e. M_{0t} = M_{s0} = 0 P -a.s..) is a strong martingale if E (M (t_{1}s_{1}, t_{2}s_{2}] |𝔉_{t_{1}s_{1}}^{1}) = 0 for any (t_{1}s_{1}, t_{2}s_{2}] ⊂ R_{11}.

9. For t_{1} < t_{2} and s_{1} < s_{2}, let ϕ ∈ L be characteristic if ϕ (t, s) = f (ω) 1_{(t_{1}s_{1},t_{2}s_{2}]} with f ∈ 𝔉_{t_{1}s_{1}}^{1}.

10. Let ϕ ∈ L be simple if it is a linear combination of characteristic functions.

11. For characteristic processes ϕ_{ts} (g) = f (g) 1_{(t_{1}s_{1},t_{2}s_{2}]}, define the integral

∫ b_{dtds}^{A}ϕ_{ts} ≡ f (g) b ((t_{1}s_{1}, t_{2}s_{2}]) .

We can do this because (t_{1}s_{1}, t_{2}s_{2}] ∩ R_{11} is a rectangle of the form (τσ, ts].

12. Extend the definition of ∫ b_{dtdσ}^{A} to simple functions by linearity.

We shall need the following results of Cairoli & Walsh which we state without proof. They are to be found in [7].

Theorem 6.4 (Cairoli & Walsh [7]). 1. For any simple ϕ ∈ L,

E |∫ b_{dtds}^{A}ϕ_{ts} (g)|^{2} = E |∫_{0}^{1} ∫_{0}^{1} ϕ_{ts} (g) dtds|^{2} |A|_{𝔎}^{2} .

2. ∫ b_{dtds}^{A} provides an isometry between L and L^{2} (Ω) and we can extend the definition of ∫ b_{dtds}^{A} to all of L via this isometry.

3. For any ϕ ∈ L, the process

M_{ts} ≡ ∫_{R_{ts}} b_{dτdσ}^{A}ϕ_{τσ} ≡ ∫ b_{dτdσ}^{A}ϕ_{τσ}1_{R_{ts}} (τ, σ)

is a strong martingale.

4. For any strong martingale M and for any p ≥ 1, we have

E sup_{τ≤t} sup_{σ≤s} |M_{τσ}|^{p} ≤ (\frac{p}{p − 1})^{2p} sup_{τ≤t} sup_{σ≤s} E |M_{τσ}|^{p} .

6.1.2. The Norm H^{2}(s).

Definition 6.5 (The Hilbert-Schmidt norm ∥·∥_{HS} on M_{m} (ℝ)). Let M_{m} (ℝ) and GL_{m} (ℝ) be as in Remark 2.18. The Hilbert-Schmidt norm ∥·∥_{HS} of a matrix k ∈ M_{m} (ℝ) with i, j-entries k_{ij} is given by

∥k∥_{HS} = (∑_{i=1}^{m} ∑_{j=1}^{m} k_{ij}^{2})^{1/2} .

[Note:- By the equivalence of norms on a linear space in finite dimensions, ∥·∥_{HS}- convergence on 𝔎 is the same as ⟨·, ·⟩_{𝔎}-convergence on 𝔎.]

The next couple of definitions have been adapted for our purposes from Chap. IV of Protter, [30].

Definition 6.6 (Very Special Semimartingales). A semimartingale in the sense of Definition 2.24 is very special if it can be written as M + ∫ ν(σ)dσ where M is a mean-zero martingale and ν is an adapted process. [Such a decomposition is always unique see Theorem 18, Chap III of [30]]. An ℝ^{d}-valued semimartingale is very special if its individual coordinates are very special. In particular, viewing M_{m} (ℝ) as ℝ^{m×m} we have a definition for very special M_{m} (ℝ)-valued semimartingales.

Definition 6.7 (The norm H^{2} (s) on M_{m} (ℝ)-semimartingales). Let M_{m} (ℝ) denote m×m matrices as in Definition 6.5. By Remark 2.18 K ⊂ GL_{m} (ℝ) ⊂ M_{m} (ℝ). Let R be a very special M_{m} (ℝ)-semimartingale which is Doob-decomposable as M + ∫ ν(σ)dσ.

∥R∥_{H^{2}(s)} ≡ ∥M_{s}∥_{L^{2}} + ∥∫_{0}^{s} ∥ν(σ)∥_{HS} dσ∥_{L^{2}}

= (E ∥M_{s}∥_{HS}^{2})^{\frac{1}{2}} + (E (∫_{0}^{s} ∥ν(σ)∥_{HS} dσ)^{2})^{\frac{1}{2}} .

Theorem 6.8 (Closure of very special semimartingales under ∥·∥_{H^{2}(s)}). Let H^{2}(s) be the space of all very special M_{m} (ℝ)-semimartingales R so that ∥R∥_{H^{2}(s)} < ∞. Then H^{2}(s) is a Banach space.

Proof. Clearly (H^{2}(s), ∥·∥_{H^{2}(s)}) is a normed vector space. It will suffice to check

completeness. Let R_{n} = M_{n} + ∫ ν_{n} (u) du be Cauchy. Then M_{n} is Cauchy in L^{2} and so converges to some martingale M. ∫ ν_{n} (u) du is also Cauchy in ∥·∥_{H^{2}(s)}. To show it converges, it will suffice to show a subsequence converges and so without losing generality we assume that

∞ >  \sqrt[∑_{n}]{E (∫_{0}^{s} ∥ν_{n+1}(u) − ν_{n}(u)∥_{HS} du)^{2}}

> E ∫_{0}^{s} ∑_{n} ∥ν_{n+1}(u) − ν_{n}(u)∥_{HS} du

Thus on a set A ⊂ Ω of measure 1, we see that the expression

∫_{0}^{s} ∑_{n} ∥ν_{n+1}(u) − ν_{n}(u)∥_{HS} du < ∞.

Thus on A, ∑_{n} ∥ν_{n+1}(u) − ν_{n}(u)∥_{HS} < ∞ a.s.-du. Thus on A there exists some random variable ν (u) so that, ν_{n}(u) → ν (u) a.s.-du. ν_{n} is adapted so ν is adapted as well. hence we are done.

6.1.3. The approximation scheme.

Definition 6.9. We will proceed to define some basic terms we will need to state the main Theorem of this section. Let g be the L (K)-valued Brownian motion from Definition 2.22. Let X be the 𝔎-valued Brownian bridge sheet given by the Fisk-Stratonowicz integral X_{ts} = ∫_{0}^{t} g_{τs}^{−1}g_{δτs} in Theorem 2.25. Then:-

1. ℙ a partition {0 = t_{0} < ... < t_{n} = T }.

2.

X^{ℙ}(t, s) ≡ X_{t_{i−1}s} \frac{t_{i} − t}{t_{i} − t_{i−1}} + X_{t_{i}s} \frac{t − t_{i−1}}{t_{i} − t_{i−1}} ∀t ∈ (t_{i−1}, t_{i}].

3. Δ_{j}X(s) ≡ X_{t_{j}s} − X_{t_{j−1}s} and Δ_{j}X^{A}(s) ≡ ⟨Δ_{j}X(s), A⟩_{𝔎} for any A ∈ 𝔎.

4. g^{ℙ}(t, s) be defined to be the solution of

∂_{t}g^{ℙ}(t, s) = g^{ℙ}(t, s)∂_{t}X^{ℙ}(t, s) and g^{ℙ}(0, s) = 1.

Observe that for any t ∈ (t_{i−1}, t_{i}] the this equation reduces to

∂_{t}g^{ℙ}(t, s) = g^{ℙ}(t, s)\frac{△_{i}X(s)}{△_{i}t}.

5. M_{m} (ℝ) be the set of all m × m matrices (see Remark 2.18).

6. Define G : GL_{m} (ℝ) → GL_{m} (ℝ) by setting G (A) ≡ A^{−1}. Recall from Remark 2.18 that GL_{m} (ℝ) denotes invertible m × m matrices.

7. F : M_{m} (ℝ) → M_{m} (ℝ) be the exponential map

F : A → ∑ A^{n}/n!.

Notice that for any A ∈ 𝔎, we have exp A = F (A) where exp : 𝔎 → K is the intrinsic exponential map on K.

8. y_{i} ≡ g^{ℙ}(t_{i}, s) = F (△_{1}X(s)) ⋯ F (△_{i}X(s)) .

9. B^{ℙ}(T, s) ≡ ∫_{0}^{s} g^{ℙ}(T, δs)g^{ℙ}(T, s)^{−1}.

Lemma 6.10. Recall that K ⊂ GL_{m} (ℝ) as in Remark 2.18. Let F and G be transformations of GL_{m} (ℝ) as in Definition 6.9. Then the following relations hold where A ∈ U and B, C ∈ 𝔎:-

1.

F′(A)B = ∫_{0}^{1} F [(1 − τ)A] BF [τA] dτ. (6.1)

2.

F′′(A)B ⊗ C
= ∫_{0}^{1} dτ ∫_{0}^{1}(1 − u)F [(1 − τ) (1 − u) A]
× CF [τ(1 − u)A] BF [uA] du
+ ∫_{0}^{1} dτ ∫_{0}^{1} uF [(1 − u)A]

× BF [(1 − τ)uA] CF [τuA] du. (6.2)

3.

G′(A)B = −A^{−1}BA^{−1}. (6.3)

4.

G′′(A)B ⊗ C = A^{−1}BA^{−1}CA^{−1} + A^{−1}CA^{−1}BA^{−1}. (6.4)

5.

sup_{A∈𝔎} ∥F′(A)B∥_{HS} ≤ Const ∥B∥_{HS} . (6.5)

6.

sup_{A∈𝔎} ∥F^{(n)}(A)B_{1} ⊗ ⋯ ⊗ B_{n}∥_{HS} ≤ Const ∥B_{1}∥_{HS} ⋯ ∥B_{n}∥_{HS} . (6.6)

Proof. See Lemma 8.8 in the appendix.

6.2. The Main Theorem .

Theorem 6.11 (Semimartingale properties of g_{T ·}). Let g be a L (K)-valued Brownian motion. Then:-

1. s → g_{T s} is a K-valued 𝔉_{T s}-semimartingale.[Note:-In Remark 6.1 we have already reached the conclusion that g_{T ·} was a semimartingale. We provide another independent proof since this fact is an easy consequence of our computation.]

2.

∫_{0}^{s} g_{T δσ}g_{T σ}^{−1} = ∫_{R_{T s}} Ad_{g_{tσ}}b_{dtdσ} − ∫_{0}^{s} \frac{dσ}{1 − σ} ∫_{0}^{T} Ad_{g_{tσ}}X_{dtσ},

where the expression ∫_{R_{T s}} Ad_{g_{tσ}}b_{dtdσ} is defined as in Theorem 6.4.

The proof of this Theorem will be given after the proof of Theorem 6.17.

Remark 6.12. (Theorem 6.11 is reasonable)g satisfies

g_{δts} = g_{ts}X_{δts} with g_{0s} = e, (6.7) where X_{·s} is the Brownian bridge sheet from Theorem 2.25. By Theorem 3.19, there is a Brownian sheet b on 𝔎 so that

X_{ts} = b_{ts} − ∫_{0}^{s} b_{tσ} \frac{(1 − s)}{(1 − σ)^{2}} dσ. (6.8)

If we replace X by b in Eq. (6.7), then Lemma 3.9 shows that s → g_{T s} would be a K-valued Brownian motion with variance T and hence ∫_{0}^{·} g_{T δs}g_{T s}^{−1} would be a 𝔎-valued Brownian motion with variance T. In reality, because X_{t·} contains an extra finite-variation term, it turns out that the law of Y_{T ·} is equivalent (but not equal) to the law of a Brownian motion on 𝔎.

Define

Y_{T s} ≡ ∫_{R_{T s}} Ad_{g_{tσ}}b_{dtdσ} − ∫_{0}^{s} \frac{dσ}{1 − σ} ∫_{0}^{T} Ad_{g_{tσ}}X_{dtσ}. (6.9)

In the proof of Theorem 6.11 we will show that g_{T δs} = Y_{T δs}g_{T s} with Y_{T 0} = 0. Before we do that we shall need to state a few results.

Proposition 6.13 (Semimartingale decomposition of B_{T ·}^{ℙ}). As in Definition 6.9 let ℙ be a partition of [0.T] and let

B_{T}^{ℙ} (s) ≡ B^{ℙ}(T, s) = ∫_{0}^{s} g^{ℙ}(T, δs)g^{ℙ}(T, s)^{−1}.

Define

(6.10)  M_{T}^{ℙ}(s) ≡ ∫_{0}^{s} ∑_{i=1}^{n_{ℙ}} y_{i−1} (σ) (F′(Δ_{i}X)dΔ_{i}b(σ)) y_{i}^{−1} (σ) ,

(6.11)  ν_{1}^{ℙ}(T, s) ≡ −\frac{1}{1 − s} ∑_{i=1}^{n_{ℙ}} Ad_{y_{i−1}}Δ_{i}X(s),

(6.12)  ν_{2}^{ℙ}(T, s) ≡ ∑_{i=1}^{n_{ℙ}} ∑_{A} \frac{(Δ_{i}t)}{2}y_{i−1} (F′′(Δ_{i}X)A^{⊗2}) y_{i}^{−1}

− ∑_{i=1}^{n_{ℙ}} ∑_{A} \frac{(Δ_{i}t)}{2} (y_{i−1} (F′(Δ_{i}X)A) y_{i}^{−1})^{2} .

Then

B_{T}^{ℙ} (s) = M_{T}^{ℙ}(s) + ∫_{0}^{s} ν_{1}^{ℙ}(T, σ)dσ + ∫_{0}^{s} ν_{2}^{ℙ}(T, σ)dσ. (6.13)

This Proposition is proved in subsection 6.4.

Remark 6.14 (Idea of the proof of Theorem 6.11). Given Theorem 6.13 we can indicate the idea of the proof of Theorem 6.11. Roughly speaking we have the following approximations

ν_{2}^{ℙ}(T, s) ≅ ∑_{i=1}^{n_{ℙ}} \frac{(Δ_{i}t)}{2}y_{i−1} (F′′(Δ_{i}X) ∑_{A} A^{⊗2}) y_{i−1}^{−1} − ∑_{i=1}^{n_{ℙ}} ∑_{A} \frac{(Δ_{i}t)}{2} (y_{i−1} (F′(Δ_{i}X)A) y_{i−1}^{−1})^{2}

= ∑_{i=1}^{n_{ℙ}} ∑_{A} \frac{(Δ_{i}t)}{2}y_{i−1} (F′′(Δ_{i}X)A^{⊗2} − (F′(Δ_{i}X)A)^{2}) y_{i−1}^{−1}

≅ ∑_{i=1}^{n_{ℙ}} ∑_{A} \frac{(Δ_{i}t)}{2}y_{i−1} (F′′(0)A^{⊗2} − (F′(0)A)^{2} + O(Δ_{i}X)) y_{i−1}^{−1} ≅ 0.

Also one expects

M_{T}^{ℙ}(s) ≡ ∫_{0}^{s} ∑_{i=1}^{n_{ℙ}} y_{i−1} (σ) (F′(Δ_{i}X)dΔ_{i}b(σ)) y_{i}^{−1} (σ) → ∫_{0}^{s} ∫_{0}^{T} Ad_{g(t,σ)}b(dt, dσ)

and

ν_{1}^{ℙ}(T, s) ≡ −\frac{1}{1 − s} ∑_{i=1}^{n_{ℙ}} Ad_{y_{i−1}}Δ_{i}X(s) → −\frac{1}{1 − s} ∫_{0}^{T} Ad_{g(t,s)}X(dt, s)

as |ℙ| → 0.

Lemma 6.15. Let z < 1. Let α ∈ [−1, 1]. Then there exists a sequence of partitions {ℙ_{r}^{z}} of [0, T], depending only on T and z with the following properties:-

1. |ℙ_{r}^{z}| ↓ 0 as r → ∞

2.

sup_{{ℙ_{r}^{z}}} sup_{σ∈[0,z]} ∑_{i=1}^{n_{ℙ^{z}_{r}}} ∥F (αΔ_{i}X(σ)) − 1∥_{HS}^{2} < ∞ P -a.s.

3. As r → ∞,

∥ sup_{t∈[0,T]} ∥g^{ℙ_{r}^{z}}(t, s) − g_{ts}∥_{HS}∥_{L^{p}} → 0 ∀p ∈ [1, ∞) , T < ∞.

This result is proved in subsection 6.5.

Theorem 6.16. Let B_{T ·}^{ℙ_{r}^{z}} be the approximation to ∫_{0}^{·} g_{T δs}g_{T s}^{−1} as in Definition 6.9. Let M_{T ·}^{ℙ_{r}^{z}} be the martingale part of B_{T ·}^{ℙ_{r}^{z}} as in Proposition 6.13. Let M_{T ·} ≡ ∫_{R_{T ·}} Ad_{g_{ts}}b_{dtds} be the martingale part of Y_{T ·} as in the proof of Theorem 6.11. Then as r → ∞ the expression

∫_{0}^{·} M_{T dσ}^{ℙ_{r}^{z}}M_{T dσ}^{ℙ_{r}^{z}}g_{T σ}^{ℙ_{r}^{z}} → ∫_{0}^{·} M_{T dσ}M_{T dσ}g_{T σ} in H^{2} (z) .

Theorem 6.17. Let t ↦ g_{t·} be the canonical L (K)-valued Brownian motion from Definition 2.22. Let Y_{T ·} be as in the proof of Theorem 6.11. Let B_{T ·}^{ℙ_{r}^{z}} be the approximation to ∫_{0}^{·} g_{T δs}g_{T s}^{−1} as in Definition 6.9. Then as r → ∞, we have

∫_{0}^{·} B_{T dσ}^{ℙ_{r}^{z}}g_{T σ}^{ℙ_{r}^{z}} → ∫_{0}^{·} Y_{T dσ}g_{T σ} in H^{2} (z) .

Proof of Theorem 6.11. Since σ → g_{T σ} is bounded and continuous, the integral

∫_{0}^{·} Y_{T dσ}g_{T σ} + \frac{1}{2}Y_{T dσ}Y_{T dσ}g_{T σ}

is well-defined. We have only to show that for any z < 1,

g_{T s}^{ℙ_{r}^{z}} → 1 + ∫_{0}^{s} Y_{T dσ}g_{T σ} + \frac{1}{2}Y_{T dσ}Y_{T dσ}g_{T σ} in H^{2}(z) as r → ∞.

If this is done, Lemma 6.15 will imply that

g_{T s} = 1 + ∫_{0}^{s} Y_{T dσ}g_{T σ} + \frac{1}{2}Y_{T dσ}Y_{T dσ}g_{T σ}.

ON THE EQUIVALENCE OF MEASURES ON LOOP SPACE 55

This will mean that g_{T ·} is a semimartingale and that

Y_{T δs}g_{T s} = Y_{T ds}g_{T s} + \frac{1}{2}Y_{T ds}g_{T ds}
= Y_{T ds}g_{T s} + \frac{1}{2}Y_{T ds}Y_{T ds}g_{T s}
= g_{T δs}.

This will prove the Theorem completely.

So will suffice to prove, for any z < 1,

g_{T s}^{ℙ_{r}^{z}} → 1 + ∫_{0}^{s} Y_{T dσ}g_{T σ} + \frac{1}{2}Y_{T dσ}Y_{T dσ}g_{T σ} in H^{2}(z) as r → ∞.

Doob-decompose Y_{T s} as M_{T s}^{ℙ_{r}^{z}} + ∫_{0}^{s} ν^{ℙ_{r}^{z}}(T, σ)dσ. From Definition 6.9, notice that g_{T δs}^{ℙ_{r}^{z}} solves

g_{T δs}^{ℙ_{r}^{z}} = B_{T δs}^{ℙ_{r}^{z}}g_{T s}^{ℙ_{r}^{z}} with g_{T 0}^{ℙ_{r}^{z}} = 1.

Now let M_{T ·}^{ℙ_{r}^{z}} be the martingale part of M_{T ·}^{ℙ_{r}^{z}} and let J denote

∥g_{T ·}^{ℙ_{r}^{z}} − (1 + ∫_{0}^{·} Y_{T dσ}g_{T σ} + \frac{1}{2}Y_{T dσ}Y_{T dσ}g_{T σ})∥_{H^{2}(z)} .

We see that

J = ∥∫_{0}^{·} (B_{T dσ}^{ℙ_{r}^{z}}g_{T σ}^{ℙ_{r}^{z}} − Y_{T dσ}g_{T σ}) + \frac{1}{2} (B_{T dσ}^{ℙ_{r}^{z}}B_{T dσ}^{ℙ_{r}^{z}}g_{T σ}^{ℙ_{r}^{z}} − Y_{T dσ}Y_{T dσ}g_{T σ})∥_{H^{2}(z)}
= ∥∫_{0}^{·} (B_{T dσ}^{ℙ_{r}^{z}}g_{T σ}^{ℙ_{r}^{z}} − Y_{T dσ}g_{T σ}) + \frac{1}{2} (M_{T dσ}^{ℙ_{r}^{z}}M_{T dσ}^{ℙ_{r}^{z}}g_{T σ}^{ℙ_{r}^{z}} − Y_{T dσ}Y_{T dσ}g_{T σ})∥_{H^{2}(z)}

By Theorem 6.16,

∫_{0}^{·} M_{T dσ}^{ℙ_{r}^{z}}M_{T dσ}^{ℙ_{r}^{z}}g_{T σ}^{ℙ_{r}^{z}} → ∫_{0}^{·} M_{T dσ}M_{T dσ}g_{T σ} in H^{2} (z) .

By Theorem 6.17,

∫_{0}^{·} B_{T dσ}^{ℙ_{r}^{z}}g_{T σ}^{ℙ_{r}^{z}} → ∫_{0}^{·} Y_{T dσ}g_{T σ} in H^{2} (z) .

Hence we are done. Remark 6.18. By Eq. (3.5),

X_{ts} = b_{ts} − ∫_{0}^{s} b_{tσ} \frac{(1 − s)}{(1 − σ)^{2}} dσ.

Computing informally with Ito’s Lemma and using Eq. (6.9),

X_{ts} = b_{ts} − ∫_{0}^{s} b_{tσ} \frac{(1 − s)}{(1 − σ)^{2}} dσ.

X_{dts} = b_{dts} − ∫_{0}^{s} b_{dtσ} \frac{(1 − s)}{(1 − σ)^{2}} dσ.

X_{dtds} = b_{dtds} − \frac{b_{dts}ds}{(1 − s)} + ∫_{0}^{s} b_{dtσ} \frac{1}{(1 − σ)^{2}} dσds
= b_{dtds} − \frac{X_{dts}ds}{(1 − s)}.

∫_{R_{T s}} Ad_{g_{tσ}}X_{dtdσ} = ∫_{R_{T s}} Ad_{g_{tσ}}b_{dtdσ} − ∫_{0}^{s} \frac{dσ}{(1 − σ)} ∫_{0}^{T} Ad_{g_{tσ}}X_{dtσ}.

∫_{R_{T s}} Ad_{g_{tσ}}X_{dtdσ} = Y_{T s}.

Thus we see that in spirit, Y_{T s} equals ∫_{R_{T s}} Ad_{g_{tσ}}X_{dtdσ}.

In future, define ∫_{R_{T s}} Ad_{g_{tσ}}X_{dtdσ} = Y_{T s}.

6.3. Proof of Theorems 6.16 and 6.17. We will need the following three Propositions (in addition to Proposition 6.13) in the proof of Theorems 6.16 and 6.17.

Proposition 6.19. Let z < 1 and let {ℙ_{r}^{z}} be the sequence of partitions from

Lemma 6.15 and let M_{T}^{ℙ_{r}^{z}} (·) be the martingale part of B_{T ·}^{ℙ_{r}^{z}} = ∫_{0}^{s} g_{T δσ}^{ℙ_{r}^{z}} (g_{T σ}^{ℙ_{r}^{z}})^{−1}.

Then M_{T}^{ℙ_{r}^{z}} (s) converges in L^{2} as r → ∞ to ∫_{R_{T s}} Ad_{g_{τσ}}b_{dτdσ}. Furthermore the process s → ∫_{R_{T s}} Ad_{g_{τσ}}b_{dτdσ} is a Brownian motion on 𝔎 with variance T.

Proposition 6.20. Let z < 1. Let {ℙ_{r}^{z}} be the sequence of partitions in Lemma 6.15. Let ν_{2}^{ℙ_{r}^{z}}(T, ·) be as in Proposition 6.13. Let convergence in H^{2} (z) be defined as in Definition 6.7. Then

∫_{0}^{·} ν_{2}^{ℙ_{r}^{z}}(T, σ)dσ → 0 as |ℙ_{r}^{z}| ↓ 0 in H^{2}(z).

Proposition 6.21. Let z < 1 and let convergence in H^{2} (z) be as in Definition 6.7. ν_{1}^{ℙ_{r}^{z}}(T, ·) be as in Proposition 6.13. Then as r → ∞ we have.

∫_{0}^{·} ν_{1}^{ℙ_{r}^{z}}(T, σ)dσ → − ∫_{0}^{·} \frac{dσ}{1 − σ} ∫_{0}^{T} Ad_{g_{tσ}}X_{dtσ} in H^{2}(z).

Proof of Theorem 6.16. By Eq. (6.10)

M_{T}^{ℙ_{r}^{z}}(s) ≡ ∫_{0}^{s} ∑_{i=1}^{n_{ℙ^{z}_{r}}} y_{i−1} (σ) (F′(Δ_{i}X)dΔ_{i}b(σ)) y_{i}^{−1} (σ) .

which implies

M_{T ds}^{ℙ_{r}^{z}}M_{T ds}^{ℙ_{r}^{z}}

= ∑_{i,j=1}^{n_{ℙ^{z}_{r}}} y_{i−1} (s) (F′(Δ_{i}X)Δ_{i}b(ds)) y_{i}^{−1} (s) y_{j−1} (s) (F′(Δ_{j}X)Δ_{j}b(ds)) y_{j}^{−1} (s)

= ∑_{A} ∑_{i=1}^{n_{ℙ^{z}_{r}}} y_{i−1} (s) (F′(Δ_{i}X)A) (y_{i}^{−1}y_{i−1}) (s) (F′(Δ_{i}X)A) y_{i}^{−1} (s) Δ_{i}tds.

By Definition 6.9, y_{i} = y_{i−1}F (△_{i}X(s)), so we have y_{i}^{−1}y_{i−1} = F (−△_{i}X(s)). Thus

M_{T ds}^{ℙ_{r}^{z}}M_{T ds}^{ℙ_{r}^{z}} = ∑_{A} ∑_{i=1} Ad_{y_{i−1}} [(F′(Δ_{i}X)A) F (−△_{i}X)]^{2} Δ_{i}tds. (6.14)

Also, by Theorem 6.19,

M_{T ds}M_{T ds} = T ds ∑_{A} A^{2} = ds ∑_{A} ∑_{i=1}^{n_{ℙ^{z}_{r}}} (Ad_{y_{i−1}}A^{2}) Δ_{i}t. (6.15)

Then using Equations (6.14) and (6.15) we have

|| ∫_{0}^{·}M_{T dσ}^{ℙ_{r}^{z}}M_{T dσ}^{ℙ_{r}^{z}}g_{T σ}^{ℙ_{r}^{z}} − M_{T dσ}M_{T dσ}g_{T σ}||_{H^{2}(z)}

= ∥ \begin{matrix}∫_{0}^{·} (M_{T dσ}^{ℙ_{r}^{z}}M_{T dσ}^{ℙ_{r}^{z}} − M_{T dσ}M_{T dσ}) g_{T σ}^{ℙ_{r}^{z}} \\ + ∫_{0}^{·} M_{T dσ}M_{T dσ} (g_{T σ}^{ℙ_{r}^{z}} − g_{T σ}) \end{matrix} ∥_{H^{2}(z)}

≤ ∥∫_{0}^{·} (M_{T dσ}^{ℙ_{r}^{z}}M_{T dσ}^{ℙ_{r}^{z}} − M_{T dσ}M_{T dσ}) g_{T σ}^{ℙ_{r}^{z}} ∥_{H^{2}(z)}

+ ∥∫_{0}^{·} M_{T dσ}M_{T dσ} (g_{T σ}^{ℙ_{r}^{z}} − g_{T σ})∥_{H^{2}(z)}

= J_{1} + J_{2}.

Using the definition of ∥·∥_{H^{2}(z)} (see Definition 6.7):

J_{2}^{2} = ∥∫_{0}^{·} M_{T dσ}M_{T dσ} (g_{T σ}^{ℙ_{r}^{z}} − g_{T σ})∥_{H^{2}(z)}^{2}

= ∥∫_{0}^{·} T dσ (∑_{A} A^{2}) (g_{T σ}^{ℙ_{r}^{z}} − g_{T σ})∥_{H^{2}(z)}^{2}

= E (∫_{0}^{z} T dσ ∥(∑_{A} A^{2}) (g_{T σ}^{ℙ_{r}^{z}} − g_{T σ})∥_{HS})^{2}

≤ CE (∫_{0}^{z} T dσ ∥g_{T σ}^{ℙ_{r}^{z}} − g_{T σ}∥_{HS})^{2} ,

which vanishes by dominated convergence and Lemma 6.15.

Again using Definition 6.7:

J_{1}^{2} = ∥∫_{0}^{·} (M_{T dσ}^{ℙ_{r}^{z}}M_{T dσ}^{ℙ_{r}^{z}} − M_{T dσ}M_{T dσ}) g_{T σ}^{ℙ_{r}^{z}} ∥_{H^{2}(z)}^{2}

= ∥∫_{0}^{·} dσ ∑_{A} ∑_{i=1}^{n_{ℙ^{z}_{r}}} Δ_{i}tAd_{y_{i−1}} ([(F′(Δ_{i}X)A) F (−△_{i}X)]^{2} − A^{2}) g_{T σ}^{ℙ_{r}^{z}} ∥_{H^{2}(z)}^{2}

= E [∫_{0}^{z} dσ ∥∑_{A} ∑_{i=1}^{n_{ℙ^{z}_{r}}} Δ_{i}tAd_{y_{i−1}} ([(F′(Δ_{i}X)A) F (−△_{i}X)]^{2} − A^{2}) g_{T σ}^{ℙ_{r}^{z}} ∥]^{2}

≤ CE (∫_{0}^{z} dσ ∑_{A} ∑_{i=1}^{n_{ℙ^{z}_{r}}} Δ_{i}t ∥[(F′(Δ_{i}X)A) F (−△_{i}X)]^{2} − A^{2}∥_{HS})^{2} .

By Lemma 6.10, F′(Δ_{i}X)A is bounded in ∥·∥_{HS}. Thus ∥(F′(Δ_{i}X)A) F (−△_{i}X)∥_{HS} and ∥A∥_{HS} are bounded. Observing this and decomposing

[(F′(Δ_{i}X)A)F (−△_{i}X)]^{2} − A^{2}

= (F′(Δ_{i}X)A) F (−△_{i}X) [(F′(Δ_{i}X)A) F (−△_{i}X) − A]

+ [(F′(Δ_{i}X)A) F (−△_{i}X) − A] A, we have

J_{1}^{2} ≤ CE (∫_{0}^{z} dσ ∑_{i,A} Δ_{i}t ∥(F′(Δ_{i}X)A) exp (−△_{i}X) − A∥_{HS})^{2} .

The integrand in this last expression is bounded and hence J_{1}^{2} vanishes by the dominated convergence Theorem.

Proof of Theorem 6.17. Doob decompose Y_{T ·} as the sum of its martingale part M_{T ·} and its bounded variation part ∫_{0}^{·} ν (σ) ds. Then, from Theorem 6.11 we have M_{T ·} ≡ ∫_{R_{T ·}} Ad_{g_{ts}}b_{dtds} and ν (σ) = −\frac{1}{1−σ} ∫_{0}^{T} Ad_{g_{tσ}}X_{dtσ}.

By the definition of ∥·∥_{H^{2}(z)} (see Definition 6.7) and the expression for B_{T s}^{ℙ_{r}^{z}} in Eq. 6.13

|| ∫_{0}^{·} B_{T dσ}^{ℙ_{r}^{z}}g_{T σ}^{ℙ_{r}^{z}} − Y_{T dσ}g_{T σ}||_{H^{2}(z)}
≤ ∥∫_{0}^{·} (B_{T dσ}^{ℙ_{r}^{z}} − Y_{T dσ}) g_{T σ}^{ℙ_{r}^{z}} ∥_{H^{2}(z)} + ∥∫_{0}^{·} Y_{T dσ} (g_{T σ}^{ℙ_{r}^{z}} − g_{T σ})∥_{H^{2}(z)}

= ∥∫_{0}^{z} (M_{T dσ}^{ℙ_{r}^{z}} − M_{T dσ}) g_{T σ}^{ℙ_{r}^{z}} ∥_{L^{2}}
+ ∥∫_{0}^{·} dσ (ν_{1}^{ℙ_{r}^{z}} (T, σ) + ν_{2}^{ℙ_{r}^{z}} (T, σ) − ν_{T σ}) g_{T σ}^{ℙ_{r}^{z}} ∥_{H^{2}(z)}

+ ∥∫_{0}^{z} M_{T dσ} (g_{T σ}^{ℙ_{r}^{z}} − g_{T σ})∥_{L^{2}} + ∥∫_{0}^{·} ν (T, σ) (g_{T σ}^{ℙ_{r}^{z}} − g_{T σ}) dσ∥_{H^{2}(z)}
= I_{1} + I_{2} + I_{3} + I_{4}.

I_{1}^{2} = E ∥∫_{0}^{z} (M_{T dσ}^{ℙ_{r}^{z}} − M_{T dσ}) g_{T σ}^{ℙ_{r}^{z}} ∥_{HS}^{2}

= E ∥∑_{A} ∫_{0}^{s} ⟨M_{T dσ}^{ℙ_{r}^{z}} − M_{T dσ}, A⟩_{𝔎} Ag_{T σ}^{ℙ_{r}^{z}} ∥_{HS}^{2}

≤ dim 𝔎 ∑_{A} E ∥∫_{0}^{s} Ag_{T σ}^{ℙ_{r}^{z}} ⟨M_{T dσ}^{ℙ_{r}^{z}} − M_{T dσ}, A⟩_{𝔎}∥_{HS}^{2}

= dim 𝔎 ∑_{A} E ∫_{0}^{s} ∥Ag_{T σ}^{ℙ_{r}^{z}} ∥_{HS}^{2} ⟨M_{T dσ}^{ℙ_{r}^{z}} − M_{T dσ}, A⟩_{𝔎}^{2}

≤ C ∑_{A} E ∫_{0}^{s} ⟨M_{T dσ}^{ℙ_{r}^{z}} − M_{T dσ}, A⟩_{𝔎}^{2}

= CE ∥M_{T dσ}^{ℙ_{r}^{z}} − M_{T dσ}∥_{𝔎}^{2} → 0 as r → ∞,

ON THE EQUIVALENCE OF MEASURES ON LOOP SPACE 59

by Theorem 6.19 and the equivalence of norms on finite-dimensional spaces.

I_{2}^{2} = ∥∫_{0}^{·} dσ (ν_{1}^{ℙ_{r}^{z}} (T, σ) + ν_{2}^{ℙ_{r}^{z}} (T, σ) − ν_{T σ}) g_{T σ}^{ℙ_{r}^{z}} ∥_{H^{2}(z)}^{2}
= E (∫_{0}^{z} ∥ν_{1}^{ℙ_{r}^{z}}(T, σ) − ν(T, σ) + ν_{2}^{ℙ_{r}^{z}}(T, σ)∥_{HS} ∥g_{T σ}^{ℙ_{r}^{z}} ∥_{HS} dσ)^{2}
≤ C ∥∫_{0}^{·} ν_{1}^{ℙ_{r}^{z}}(T, σ) − ν(T, σ)∥_{H^{2}(z)}^{2} + C ∥∫_{0}^{·} ν_{2}^{ℙ_{r}^{z}}(T, σ)∥_{H^{2}(z)}^{2}
→ 0 as r → ∞ by Theorems 6.20 and 6.21.
I_{3}^{2} = E ∥∫_{0}^{z} M_{T dσ} (g_{T σ}^{ℙ_{r}^{z}} − g_{T σ})∥_{L^{2}}^{2}
= E ∥∫_{0}^{z} ∑_{A} (Ag_{T σ}^{ℙ_{r}^{z}} − Ag_{T σ}) M_{T dσ}^{A}∥_{HS}^{2}
= dim 𝔎 ∑_{A} E ∥∫_{0}^{z} (Ag_{T σ}^{ℙ_{r}^{z}} − Ag_{T σ}) M_{T dσ}^{A}∥_{HS}^{2}
= dim 𝔎 ∑_{A} E ∫_{0}^{z} ∥Ag_{T σ}^{ℙ_{r}^{z}} − Ag_{T σ}∥_{HS}^{2} M_{T dσ}^{A}M_{T dσ}^{A}
= dim 𝔎 ∑_{A} E ∫_{0}^{z} ∥Ag_{T σ}^{ℙ_{r}^{z}} − Ag_{T σ}∥_{HS}^{2} T dσ
→ 0 by Dominated Convergence.
I_{4}^{2} = E (∫_{0}^{z} ∥ν(T, σ)∥_{HS} ∥g_{T σ}^{ℙ_{r}^{z}} − g_{T σ}∥_{HS} dσ)^{2}
≤ E ∫_{0}^{z} ∥ν(T, σ)∥_{HS}^{2} dσ ∫_{0}^{z} ∥g_{T σ}^{ℙ_{r}^{z}} − g_{T σ}∥_{HS} dσ
≤ \sqrt{E (∫_{0}^{z} ∥ν(T, σ)∥_{HS}^{2} dσ)^{2}}\sqrt{E (∫_{0}^{z} ∥g_{T σ}^{ℙ_{r}^{z}} − g_{T σ}∥_{HS} dσ)^{2}}
≤ \sqrt{E ∫_{0}^{z} ∥ν(T, σ)∥_{HS}^{4} dσ}\sqrt{E (∫_{0}^{z} ∥g_{T σ}^{ℙ_{r}^{z}} − g_{T σ}∥_{HS} dσ)^{2}}.

We will be done by dominated convergence if we can only show that

E ∫_{0}^{z} ∥ν(T, σ)∥_{HS}^{4} dσ < ∞.

But ν(T, σ) = −\frac{1}{1−σ} X̃_{T σ} where X̃_{·σ} ≡ ∫_{0}^{·} Ad_{g_{uσ}} X_{duσ}. Thus

E ∫_{0}^{z} ∥ν(T, σ)∥_{HS}^{4} dσ ≤ \frac{1}{(1 − z)^{4}} sup_{[0,z]} E ∥X̃_{T σ}∥_{HS}^{4}
< \frac{1}{(1 − z)^{4}} sup_{[0,z]} 3 (T σ − σ^{2})^{2} ,

60 VIKRAM K. SRIMURTHY

since X̃_{·σ} is a Brownian motion with parameter (σ − σ^{2}) by Lemma 7.10. Hence we are done. 6.4. Propositions 6.13, 6.19, 6.20, 6.21. 6.4.1. Proof of Proposition 6.13. .

y_{n} (s) = F (Δ_{1}X (s)) ⋯ F (Δ_{i}X (s))
δy_{n} (s) = ∑_{i=1}^{n_{ℙ}} y_{i−1} (s) (δF (△_{i}X (s))) y_{i}^{−1} (s) y_{n} (s)
δy_{n} (s) y_{n}^{−1} (s) = ∑_{i=1}^{n_{ℙ}} y_{i−1} (s) (δF (△_{i}X (s))) y_{i}^{−1} (s)
= ∑_{i=1}^{n_{ℙ}} Ad_{y_{i−1}} ((δF (△_{i}X)) F (−△_{i}X))
= ∑_{i=1}^{n_{ℙ}} Ad_{y_{i−1}} ((dF (△_{i}X)) F (−△_{i}X))

+\frac{1}{2} ∑_{i=1}^{n_{ℙ}} Ad_{y_{i−1}} (dF (△_{i}X) dF (−△_{i}X))

+\frac{1}{2} ∑_{i=1}^{n_{ℙ}} (dAd_{y_{i−1}}) (dF (△_{i}X) F (−△_{i}X))

= I + J + K.

Letting A run through an orthonormal basis of 𝔎 we can write dF (△_{i}X (s)) as (6.16) dF (△_{i}X) = ∑_{A} d△_{i}b^{A} (s) (F′ (△_{i}X (s)) A) + finite variation terms.

From Lemma 8.3 we can see that d△_{i}b^{A} (s) d△_{j}b^{B} (s) = δ_{ij} ⟨A, B⟩_{𝔎} (△_{i}t) ds. (6.17) From Eq. [6.16] and Eq. [6.17] above, we can conclude that

K = 0

and that

J = \frac{1}{2} ∑_{i=1}^{n_{ℙ}} Ad_{y_{i−1}} (dF (△_{i}X) d (F (△_{i}X)^{−1}))
= −\frac{1}{2} ∑_{i=1}^{n_{ℙ}} y_{i−1} (dF (△_{i}X)) F (△_{i}X)^{−1} (dF (△_{i}X)) F (△_{i}X)^{−1} y_{i−1}^{−1}
= −\frac{1}{2} ∑_{i=1}^{n_{ℙ}} (y_{i−1} (dF (△_{i}X)) y_{i}^{−1})^{2}
= −\frac{1}{2} ∑_{i=1}^{n_{ℙ}} ∑_{A} (y_{i−1} (F′ (△_{i}X) A) y_{i}^{−1})^{2} (△_{i}t) ds.

ON THE EQUIVALENCE OF MEASURES ON LOOP SPACE 61

By Ito’s Lemma, I can be computed as

I = ∑_{i=1}^{n_{ℙ}} y_{i−1} (F′(Δ_{i}X) (dΔ_{i}X) + \frac{1}{2}F′′(Δ_{i}X) (dΔ_{i}X)^{⊗2}) y_{i}^{−1}.

Thus

δy_{n} (s) y_{n}^{−1} (s) = ∑_{i=1}^{n_{ℙ}} y_{i−1} (F′(Δ_{i}X) (dΔ_{i}X) + \frac{1}{2}F′′(Δ_{i}X) (dΔ_{i}X)^{⊗2}) y_{i}^{−1}

−\frac{1}{2} ∑_{i=1}^{n_{ℙ}} ∑_{A} (y_{i−1} (F′ (△_{i}X) A) y_{i}^{−1})^{2} (△_{i}t) ds.

Using this result , Ito’s Lemma, and Eq. (3.5),

B_{T}^{ℙ} (ds) = ∑_{i=1}^{n_{ℙ}} y_{i−1} (F′(Δ_{i}X)dΔ_{i}X) y_{i}^{−1}

+\frac{1}{2} ∑_{i=1}^{n_{ℙ}} ∑_{A} (Δ_{i}t) y_{i−1} (F′′(Δ_{i}X)A^{⊗2}) y_{i}^{−1}ds

−\frac{1}{2} ∑_{i=1}^{n_{ℙ}} ∑_{A} Δ_{i}t (y_{i−1} (F′(Δ_{i}X)A) y_{i}^{−1})^{2} ds

= ∑_{i=1}^{n_{ℙ}} y_{i−1} (s) (F′(Δ_{i}X)d_{s} (Δ_{i}b (s) − ∫_{0}^{s} Δ_{i}b (σ) \frac{(1 − s)}{(1 − σ)^{2}} dσ)) y_{i}^{−1} (s)

+\frac{1}{2} ∑_{i=1}^{n_{ℙ}} ∑_{A} (Δ_{i}t) [ \begin{matrix}y_{i−1} (F′′(Δ_{i}X)A^{⊗2}) y_{i}^{−1} \\ − (y_{i−1} (F′(Δ_{i}X)A) y_{i}^{−1})^{2} \end{matrix} ] ds.

Therefore,

dB_{T}^{ℙ} (s) = ∑_{i=1}^{n_{ℙ}} y_{i−1} (F′(Δ_{i}X)Δ_{i}b(ds)) y_{i}^{−1}

− ∑_{i=1}^{n_{ℙ}} y_{i−1} [F′(Δ_{i}X) (\frac{Δ_{i}b(s)}{(1 − s)} − ∫_{0}^{s} \frac{Δ_{i}b(σ)}{(1 − σ)^{2}} dσ)] y_{i}^{−1}ds

+\frac{1}{2} ∑_{i=1}^{n_{ℙ}} ∑_{A} (Δ_{i}t) [ \begin{matrix}y_{i−1} (F′′(Δ_{i}X)A^{⊗2}) y_{i}^{−1} \\ − (y_{i−1} (F′(Δ_{i}X)A) y_{i}^{−1})^{2} \end{matrix} ] ds.

= dM_{T}^{ℙ}(s) + ν̃_{1}^{ℙ}(T, s)ds + ν_{2}^{ℙ}(T, s)ds;

where we have defined

ν̃_{1}^{ℙ}(T, s) ≡ − ∑_{i=1}^{n_{ℙ}} y_{i−1} [F′(Δ_{i}X) (\frac{Δ_{i}b(s)}{(1 − s)} − ∫_{0}^{s} \frac{Δ_{i}b(σ)}{(1 − σ)^{2}} dσ)] y_{i}^{−1}.

So to be done, we only need to show that ν_{1}^{ℙ} (T, s) = ν̃_{1}^{ℙ}(T, s).

By Eq. (3.5) of Theorem 3.19

\frac{X_{ts}}{1 − s} = \frac{b_{ts}}{1 − s} − ∫_{0}^{s} \frac{b_{tσ}dσ}{(1 − σ)^{2}} .

So simplifying, we see that

ν̃_{1}^{ℙ}(T, s) = −\frac{1}{1 − s} ∑_{i=1}^{n_{ℙ}} y_{i−1} [F′(Δ_{i}X)Δ_{i}X(s)] y_{i}^{−1}.

But again we must make the observation that F |_{𝔎} is the Lie group exponential map. Therefore we have t ↦ F (tA) satisfies F′ (tA) A = F (tA) A. Therefore we see that

ν̃_{1}^{ℙ}(T, s) = −\frac{1}{1 − s} ∑_{i=1}^{n_{ℙ}} y_{i−1} [F (Δ_{i}X)Δ_{i}X(s)] y_{i}^{−1}.

Now from Definition 6.9 observe that y_{i} (s) = y_{i−1} (s) F (△_{i}X (s)). Thus y_{i}^{−1}y_{i−1} = F (−△_{i}X (s)).

Thus we are done since

ν̃_{1}^{ℙ}(T, s) = −\frac{1}{1 − s} ∑_{i=1}^{n_{ℙ}} Ad_{y_{i−1}} [F (Δ_{i}X)Δ_{i}X(s)F (−△_{i}X (s))]

= −\frac{1}{1 − s} ∑_{i=1}^{n_{ℙ}} Ad_{y_{i−1}}Δ_{i}X(s)
= ν_{1}^{ℙ}(T, s).

6.4.2. Proof of Proposition 6.19.

Lemma 6.22. Recall that M_{T}^{ℙ_{r}^{z}}(s) is the martingale part of B_{T}^{ℙ_{r}^{z}}(s). Then

M_{T}^{ℙ_{r}^{z}}(s) “approximates” ∫_{0}^{s} ∑_{i=1}^{n_{ℙ^{z}_{r}}} y_{i−1} (dΔ_{i}b) y_{i}^{−1}.

Specifically, for any s ∈ [0, z],

∥M_{T}^{ℙ_{r}^{z}}(s) − ∫_{0}^{s} ∑_{i=1}^{n_{ℙ^{z}_{r}}} y_{i−1} (dΔ_{i}b) y_{i}^{−1}∥_{L^{2}} → 0 as r → ∞.

Proof.

Let J ≡ ∥M_{T}^{ℙ_{r}^{z}}(s) − ∫_{0}^{s} ∑_{i=1}^{n_{ℙ^{z}_{r}}} y_{i−1} (dΔ_{i}b) y_{i}^{−1}∥_{L^{2}} .

Using Eq.(6.10) yields and Lemma 8.7,

J^{2} = E ∥∫_{0}^{s} ∑_{i=1}^{n_{ℙ^{z}_{r}}} y_{i−1} (dΔ_{i}b) y_{i}^{−1} − ∫_{0}^{s} ∑_{i=1}^{n_{ℙ^{z}_{r}}} y_{i−1} (F′(Δ_{i}X)dΔ_{i}b) y_{i}^{−1}∥_{HS}^{2}

= E ∥∫_{0}^{s} ∑_{i=1}^{n_{ℙ^{z}_{r}}} ∑_{A} y_{i−1} (A − F′(Δ_{i}X)A) y_{i}^{−1}dΔ_{i}b^{A}∥_{HS}^{2} .

Applying Lemma 8.7 to this last term shows that

J^{2} = E ∑_{i,A} Δ_{i}t ∫_{0}^{s} ∥y_{i−1} (A − F′(Δ_{i}X)A) y_{i}^{−1}∥_{HS}^{2} dσ

= C ∑_{i,A} Δ_{i}t ∫_{0}^{s} dσE ∥A − F′(Δ_{i}X)A∥_{HS}^{2} .

Appealing to Eq. (6.1), the expression

∥A − F′(Δ_{i}X)A∥_{HS}^{2} = ∥∫_{0}^{1} {A − F ((1 − τ)Δ_{i}X) AF (τΔ_{i}X)} dτ∥_{HS}^{2}

is bounded because e^{αΔ_{i}X} is group valued (and hence bounded). Thus J^{2} → 0 as r → ∞ by Dominated Convergence.

Lemma 6.23. The expression

∫_{0}^{s} ∑_{i=1}^{n_{ℙ^{z}_{r}}} y_{i−1} (dΔ_{i}b) y_{i}^{−1} “approximates” ∫_{0}^{s} ∑_{i=1}^{n_{ℙ^{z}_{r}}} (Ad_{y_{i−1}}dΔ_{i}b) .

Specifically, for any s ∈ [0, z] we have as r → ∞

∥∫_{0}^{s} ∑_{i=1}^{n_{ℙ^{z}_{r}}} y_{i−1} (dΔ_{i}b) y_{i}^{−1} − ∫_{0}^{s} ∑_{i=1}^{n_{ℙ^{z}_{r}}} (Ad_{y_{i−1}}dΔ_{i}b)∥_{L^{2}} → 0.

Proof. Using Lemma 6.15,and Dominated Convergence,

lim_{r→∞} ∥∫_{0}^{s} ∑_{i=1}^{n_{ℙ^{z}_{r}}} (Ad_{y_{i−1}}dΔ_{i}b) − ∫_{0}^{s} ∑_{i=1}^{n_{ℙ^{z}_{r}}} y_{i−1} (dΔ_{i}b) y_{i}^{−1}∥_{L^{2}}^{2}

= lim_{r→∞} E ∥∫_{0}^{s} ∑_{i=1}^{n_{ℙ^{z}_{r}}} ∑_{A} Ad_{y_{i−1}} (A − AF (−Δ_{i}X)) dΔ_{i}b^{A}∥_{HS}^{2}

≤ lim_{r→∞} E ∑_{i,A} Δ_{i}t ∫_{0}^{s} ∥Ad_{y_{i−1}} (A − AF (−Δ_{i}X))∥_{HS}^{2} dσ

≤ lim_{r→∞} CE ∫_{0}^{s} ∑_{i} Δ_{i}t ∥1 − F (−Δ_{i}X)∥_{HS}^{2} dσ

= CE ∫_{0}^{s} lim_{r→∞} ∑_{i} Δ_{i}t ∥1 − F (−Δ_{i}X)∥_{HS}^{2} dσ = 0.

The expression ∑_{i} Δ_{i}t ∥1 − F (−Δ_{i}X)∥_{HS}^{2} is bounded since F (−Δ_{i}X) is K-valued and ∑_{i} Δ_{i}t = 1. So by Dominated convergence, this last term becomes

= CE ∫_{0}^{s} lim_{r→∞} ∑_{i} Δ_{i}t ∥1 − F (−Δ_{i}X)∥_{HS}^{2} dσ.

Now

lim_{r→∞} ∑_{i}Δ_{i}t ∥1 − F (−Δ_{i}X)∥_{HS}^{2}

≤ lim_{r→∞} |ℙ_{r}^{z}| sup_{r} ∑_{i} ∥1 − F (−Δ_{i}X)∥_{HS}^{2} .

By Lemma 6.15 this last expression goes to 0 in the limit as r → ∞.

Lemma 6.24. Let ℙ be a partition of [0, T], g_{tσ}^{ℙ} the approximation to g_{tσ}, and y_{i} (σ) = g_{t_{i}σ}^{ℙ} as in Definition 6.9. Then for any s ∈ [0, z],

∫_{0}^{s} ∑_{i=1}^{n_{ℙ}} (Ad_{y_{i−1}(σ)}dΔ_{i}b (σ)) = ∫_{R_{T s}} Ad_{g_{[τ]σ}^{ℙ}}b_{dτdσ}.

where [τ] ≡ t_{i−1}∀τ ∈ (t_{i−1}, t_{i}].

Proof. It will suffice to show

∫_{0}^{s} Ad_{y_{i−1}(σ)}△_{i}b (dσ) = ∫_{(t_{i−1}0,t_{i}s]} Ad_{g_{t_{i−1}σ}^{ℙ}}b_{dτdσ}.

Letting ℚ_{r} be a refining sequence of partitions of [0, s], we have, by Theorem 6.4

∫_{(t_{i−1}0,t_{i}s]} Ad_{g_{t_{i−1}σ}^{ℙ}}b_{dτdσ} = lim_{r→∞} ∑_{s_{j}∈ℚ_{r}} Ad_{g_{t_{i−1}s_{j−1}}^{ℙ}} b (t_{i−1}s_{j−1,}t_{i}s_{j}] ;

where the limit is taken in L^{2}. However,

lim_{r→∞} ∑_{s_{j}∈ℚ_{r}}Ad_{g_{t_{i−1}s_{j−1}}^{ℙ}} b (t_{i−1}s_{j−1,}t_{i}s_{j}]

= ∫_{0}^{s} Ad_{g_{t_{i−1}σ}^{ℙ}} (b_{t_{i}dσ} − b_{t_{i−1}dσ})
= ∫_{0}^{s} Ad_{y_{i−1}(σ)}△_{i}b (dσ) .

As usual, we use the fact that g^{ℙ} is bounded and the metric is Ad-invariant. So dominated convergence goes through and L^{2} convergence is justified.

Lemma 6.25. Let g_{τσ}^{ℙ_{r}^{z}} be the approximation to g_{τσ} from Definition 6.9. Then for any s ∈ [0, z],

∫_{R_{T s}} Ad_{g_{[τ]σ}^{ℙ^{z}_{r}}}b_{dτdσ} → ∫_{R_{T s}} Ad_{g_{τσ}}b_{dτdσ} in L^{2} as r → ∞.

Proof. Let

J ≡ ∥∫_{R_{T s}} Ad_{g_{[τ]σ}^{ℙ^{z}_{r}}}b_{dτdσ} − ∫_{R_{T s}} Ad_{g_{τσ}}b_{dτdσ}∥_{L^{2}}^{2} .

By Theorem 6.4,

J = E ∥∑_{A} ∫_{R_{T s}} (Ad_{g_{[τ]σ}^{ℙ^{z}_{r}}}A − Ad_{g_{τσ}}A) b_{dτdσ}^{A}∥_{HS}^{2} .

J ≤ C ∑_{A} ∑_{pq} E (∫_{R_{T s}} (Ad_{g_{[τ]σ}^{ℙ^{z}_{r}}}A − Ad_{g_{τσ}}A)_{pq} b_{dτdσ}^{A})^{2}

= C ∑_{A} ∑_{pq} E ∫_{0}^{T} dτ ∫_{0}^{s} dσ ((Ad_{g_{[τ]σ}^{ℙ^{z}_{r}}}A − Ad_{g_{τσ}}A)_{pq})^{2}

= C ∑_{A} E ∫_{0}^{T} dτ ∫_{0}^{s} dσ ∥(Ad_{g_{[τ]σ}^{ℙ^{z}_{r}}}A − Ad_{g_{τσ}}A)_{pq}∥_{HS}^{2}

≤ C ∑_{A} E ∫_{0}^{T} dτ ∫_{0}^{s} dσ ∥g_{[τ]σ}^{ℙ_{r}^{z}} − g_{τσ}∥_{HS}^{2} + ∥(g_{[τ]σ}^{ℙ_{r}^{z}})^{−1} − (g_{τσ})^{−1}∥_{HS}^{2}
→ 0 as r → ∞ by Dominated Convergence and Lemma 6.15.

Proof of Proposition 6.19. For the purposes of this proof, define the symbol ‘∽’ to mean “has the same limit in L^{2} as |ℙ_{r}^{z}| → 0”). Defining [τ] ≡ t_{i−1} ∀τ ∈ (t_{i−1}, t_{i}], for any s ∈ [0, z],

M_{T}^{ℙ_{r}^{z}}(s) ∽ ∫_{0}^{s} ∑_{i=1}^{n_{ℙ^{z}_{r}}} y_{i−1} (dΔ_{i}b) y_{i}^{−1} by Lemma 6.22;

∫_{0}^{s} ∑_{i=1}^{n_{ℙ^{z}_{r}}} y_{i−1} (dΔ_{i}b) y_{i}^{−1} ∽ ∫_{0}^{s} ∑_{i=1}^{n_{ℙ^{z}_{r}}} (Ad_{y_{i−1}}dΔ_{i}b) by Lemma 6.23;

∫_{0}^{s} ∑_{i=1}^{n_{ℙ^{z}_{r}}} (Ad_{y_{i−1}}dΔ_{i}b) = ∫_{R_{1s}} Ad_{g_{[τ]σ}^{ℙ^{z}_{r}}}b_{dτdσ} by Lemma 6.24;

∫_{R_{T s}} Ad_{g_{[τ]σ}^{ℙ^{z}_{r}}}b_{dτdσ} ∽ ∫_{R_{T s}} Ad_{g_{τσ}}b_{dτdσ} by Lemma 6.25.

Putting all this together yields

M_{T}^{ℙ_{r}^{z}}(s) ∽ ∫_{R_{T s}} Ad_{g_{τσ}}b_{dτdσ}.

We have still to show that ∫_{R_{T s}} Ad_{g_{τσ}}b_{dτdσ} is a 𝔎-valued Brownian motion with parameter T. Let J denote the process

J ≡ ∫_{0}^{·} ∑_{i=1}^{n_{ℙ^{z}_{r}}} (Ad_{y_{i−1}}dΔ_{i}b) .

By Lemmas 6.24 and 6.25

J_{s} ∽ ∫_{R_{T s}} Ad_{g_{τσ}}b_{dτdσ}.

Thus, since L^{2} limits of Brownian motions are Brownian motions, it suffices to show the process s → J_{s} is a Brownian motion.

The rest of the proof is devoted to showing that s → J_{s} is a Brownian motion. with parameter T. We shall use the notation [N]_{·} to denote the quadratic variation of a martingale N. If N is an ℝ^{d}-valued martingale then

[N]_{·} = ∑_{i,j} [N^{(i)}, N^{(j)}] e_{i} ⊗ e_{j},

where e_{i} is a basis for ℝ^{d} and [N^{(i)}, N^{(j)}] are the joint quadratic variations of the ℝ-valued martingales N^{(i)} and N^{(j)}. Let E_{pq} be the matrix with ij-entry δ_{ip}δ_{jq}. Letting {A} run through an orthonormal basis of 𝔎, we have

[J]_{s} = [∑_{i=1}^{n_{ℙ^{z}_{r}}} ∑_{A} ∫_{0}^{·} (Ad_{y_{i−1}}A) dΔ_{i}b^{A}]_{s}

= [∑_{pq} ∑_{i=1}^{n_{ℙ^{z}_{r}}} ∑_{A} (∫_{0}^{·} (Ad_{y_{i−1}}A)_{pq} dΔ_{i}b^{A}) E_{pq}]_{s}

= ∑_{i=1}^{n_{ℙ^{z}_{r}}} ∑_{A} ∑_{pq} ∑_{p′q′} (E_{pq} ⊗ E_{p′q′}) Δ_{i}t ∫_{0}^{s} (Ad_{y_{i−1}}A)_{pq} (Ad_{y_{i−1}}A)_{p′q′} dσ

= ∑_{i=1}^{n_{ℙ^{z}_{r}}} ∑_{A} Δ_{i}t ∫_{0}^{s} (Ad_{y_{i−1}}A) ⊗ (Ad_{y_{i−1}}A) dσ

= ∑_{i=1}^{n_{ℙ^{z}_{r}}} Δ_{i}t ∫_{0}^{s} (∑_{A} A ⊗ A) dσ

= (∑_{A} A^{⊗2}) T s.

Thus by Levy’s Theorem s → J_{s} is a Brownian motion with parameter t.

6.4.3. Proof of Proposition 6.20.

Lemma 6.26. The expression ∑_{ℙ_{r}^{z}} (△_{i}t) y_{i−1} (F′′ (△_{i}X) A^{⊗2}) y_{i}^{−1} is approximately the same as the expression ∑_{ℙ_{r}^{z}} (△_{i}t) Ad_{y_{i−1}}F′′ (△_{i}X) A^{⊗2}. Specifically, P -a.s. as r → ∞,

J_{r} ≡ ∥∑_{ℙ_{r}^{z}} (△_{i}t) [y_{i−1} (F′′ (△_{i}X) A^{⊗2}) y_{i}^{−1} − Ad_{y_{i−1}}F′′ (△_{i}X) A^{⊗2}]∥_{HS} → 0.

Proof. Recall y_{i} = y_{i−1}F (△_{i}X) as in Definition 6.9. Using the boundedness of the Adjoint operator (i.e. the fact that sup_{k∈K} |Ad_{k}| < ∞) gives us

J_{r} = ∥∑_{ℙ_{r}^{z}} (△_{i}t) [Ad_{y_{i−1}} (F′′ (△_{i}X) A^{⊗2})] (y_{i−1}y_{i}^{−1} − 1)∥_{HS}

≤ C ∑_{ℙ_{r}^{z}} (△_{i}t) ∥F (−△_{i}X) − 1∥_{HS}
≤ C ∑_{ℙ_{r}^{z}} (△_{i}t)^{2} ∑_{ℙ_{r}^{z}} ∥F (−△_{i}X) − 1∥_{HS}^{2}
≤ C |ℙ_{r}^{z}| sup_{r} ∑_{ℙ_{r}^{z}} ∥F (−△_{i}X) − 1∥_{HS}^{2} .

By Lemma 6.15,

sup_{r} ∑_{ℙ_{r}^{z}} ∥F (−△_{i}X) − 1∥_{HS}^{2} < ∞, P -a.s.

Hence we are done.

Lemma 6.27. The expression ∑_{ℙ_{r}^{z}} (△_{i}t) {y_{i−1} (F′ (△_{i}X) A) y_{i}^{−1}}^{2} is approximately the same as the expression ∑_{ℙ_{r}^{z}} (△_{i}t) Ad_{y_{i−1}} (F′ (△_{i}X) A)^{2}. Specifically almost surely as r → ∞, we have, P -a.s., that

J_{r} ≡ ∥∑_{ℙ_{r}^{z}} (△_{i}t) [{y_{i−1} (F′ (△_{i}X) A) y_{i}^{−1}}^{2} − Ad_{y_{i−1}} (F′ (△_{i}X) A)^{2}]∥_{HS} → 0.

Proof.

J_{r}

= ∑_{ℙ_{r}^{z}} (△_{i}t) Ad_{y_{i−1}} {(F′ (△_{i}X) A) y_{i}^{−1}y_{i−1} (F′ (△_{i}X) A) y_{i}^{−1}y_{i−1} − (F′ (△_{i}X) A)^{2}}

= ∑_{ℙ_{r}^{z}} (△_{i}t) Ad_{y_{i−1}} (F′ (△_{i}X) A)

× Ad_{y_{i−1}} [y_{i}^{−1}y_{i−1} (F′ (△_{i}X) A) y_{i}^{−1}y_{i−1} − (F′ (△_{i}X) A)]

= ∑_{ℙ_{r}^{z}} (△_{i}t) Ad_{y_{i−1}} (F′ (△_{i}X) A)

× Ad_{y_{i−1}} {y_{i}^{−1}y_{i−1} (F′ (△_{i}X) A) y_{i}^{−1}y_{i−1} − y_{i}^{−1}y_{i−1} (F′ (△_{i}X) A)}

+ ∑_{ℙ_{r}^{z}} (△_{i}t) Ad_{y_{i−1}} (F′ (△_{i}X) A)

× Ad_{y_{i−1}} {y_{i}^{−1}y_{i−1} (F′ (△_{i}X) A) − (F′ (△_{i}X) A)}

= ∑_{ℙ_{r}^{z}} (△_{i}t) Ad_{y_{i−1}} {(F′ (△_{i}X) A) y_{i}^{−1}y_{i−1} (F′ (△_{i}X) A) (y_{i}^{−1}y_{i−1} − 1)}

+ ∑_{ℙ_{r}^{z}} (△_{i}t) Ad_{y_{i−1}} {(F′ (△_{i}X) A) (y_{i}^{−1}y_{i−1} − 1) (F′ (△_{i}X) A)} .

Bringing the Hilbert-Schmidt norm within the sum, exploiting the boundedness of the Adjoint operator, and using Eq. (6.6) of Lemma 6.10; we see that the norm of this last expression is bounded above by

Const ∑_{ℙ_{r}^{z}} (△_{i}t) ∥y_{i}^{−1}y_{i−1} − 1∥_{HS}

≤ Const (∑_{ℙ_{r}^{z}} (△_{i}t)^{2} ∑_{ℙ_{r}^{z}} ∥y_{i}^{−1}y_{i−1} − 1∥_{HS}^{2})^{1/2} .

Now from Definition 6.9 we see that y_{i}^{−1}y_{i−1} = F (−△_{i}X). Now invoking Lemma 6.15 we see that this last expression vanishes in the limit.

Lemma 6.28. As usual, let F be the exponential map as in Definition 6.9. Then as r → ∞ we have, P -a.s., that

∑_{ℙ_{r}^{z}} (△_{i}t) ∥F′′ (△_{i}X) A^{⊗2} − (F′ (△_{i}X) A)^{2}∥_{HS} → 0.

Proof. Let I_{r},J_{r} be defined as follows:

I_{r} ≡ ∑_{ℙ_{r}^{z}} (△_{i}t) ∥F′′ (△_{i}X) A^{⊗2} − A^{2}∥_{HS} ;

J_{r} ≡ ∑_{ℙ_{r}^{z}} (△_{i}t) ∥A^{2} − (F′ (△_{i}X) A)^{2}∥_{HS} .

It will suffice to show that the random variables I_{r} and J_{r} vanish almost surely as r → ∞. To do this, first notice that F′ (0) A = A and that F′′ (0) A^{⊗2} = A^{2}.

The expression

I_{r} = ∑_{ℙ_{r}^{z}} (△_{i}t) ∥F′′ (△_{i}X) A^{⊗2} − F′′ (0) A^{⊗2}∥_{HS}

= ∑_{ℙ_{r}^{z}} (△_{i}t) ∥∫_{0}^{1} (F′′′ (ε△_{i}X) A ⊗ A ⊗ △_{i}X) dε∥_{HS}

≤ ∑_{ℙ_{r}^{z}} (△_{i}t) ∫_{0}^{1} ∥F′′′ (ε△_{i}X) A ⊗ A ⊗ △_{i}X∥_{HS} dε.
≤ C ∑_{ℙ_{r}^{z}} (△_{i}t) ∥△_{i}X∥_{HS} , by Lemma 6.10.

However, the expression

C ∑_{ℙ_{r}^{z}} (△_{i}t) ∥△_{i}X∥_{HS} ≤ C [sup_{r} ∑_{ℙ_{r}^{z}} ∥△_{i}X∥_{HS}^{2}] ∑_{ℙ_{r}^{z}} (△_{i}t)^{2} . (6.18)

Invoking Eq. [6.24] in the proof of Lemma 6.15, we see that the right hand side of Eq. [6.18] goes to zero as r → ∞. Thus I_{r} → 0, P -a.s., as r → ∞.

Turning now to J_{r}, we see that

J_{r} = ∑_{ℙ_{r}^{z}} (△_{i}t) ∥(F′ (△_{i}X) A)^{2} − (F′ (0) A)^{2}∥_{HS}

= ∑_{ℙ_{r}^{z}} (△_{i}t) 2 ∫_{0}^{1} ∥F′ (ε△_{i}X) A∥_{HS} ∥F′′ (ε△_{i}X) A ⊗ △_{i}X∥_{HS}
≤ C ∑_{ℙ_{r}^{z}} (△_{i}t) ∥△_{i}X∥_{HS} by Lemma 6.10.

Therefore,

J_{r} ≤ C ∑_{ℙ_{r}^{z}} (△_{i}t) ∥△_{i}X∥_{HS} → 0 by Eqs. [6.18] and [6.24].

Proof of Proposition 6.20. Let y_{i} (·) denote g_{t_{i}·}^{ℙ_{r}^{z}} as in Definition 6.9. Let F be the exponential map acting on m × m matrices as in Definition 6.9. Let A run through an orthonormal basis of 𝔎. Let

J_{r} ≡ ∥∫_{0}^{·} ν_{2}^{ℙ_{r}^{z}}(T, σ)dσ∥_{H^{2}(z)}^{2} .

Then using Hölder’s inequality and the fact that s ≤ 1 yields

J_{r} (6.19)

≤ E ∫_{0}^{z} dσ ∥∑_{ℙ_{r}^{z},A} \frac{△_{i}t}{4} [y_{i−1} (F′′ (△_{i}X) A^{⊗2}) y_{i}^{−1} − (y_{i−1} (F′ (△_{i}X) A) y_{i}^{−1})^{2}]∥_{HS}^{2} .

Notice that by invoking Eq. (6.6) of Lemma 6.10 as well as the boundedness of K, we see that the Hilbert-Schmidt norm in Eq. (6.19) is bounded. Thus we can invoke dominated convergence and so it suffices to show for fixed A that as r → ∞, we have, P -a.s., that the expression

∑_{ℙ_{r}^{z}} (△_{i}t) [y_{i−1} (F′′ (△_{i}X) A^{⊗2}) y_{i}^{−1} − (y_{i−1} (F′ (△_{i}X) A) y_{i}^{−1})^{2}] → 0.

For simplicity, we shall use “∽” to mean “has the same limit in 𝔎”. Explicitly {f_{ℙ_{r}^{z}}} ∽ {g_{ℙ_{r}^{z}}} iff ∥f_{ℙ_{r}^{z}} − g_{ℙ_{r}^{z}} ∥_{HS} → 0, P -a.s., as r → ∞. So the problem reduces to showing

∑_{ℙ_{r}^{z}} (△_{i}t) y_{i−1} (F′′ (△_{i}X) A^{⊗2}) y_{i}^{−1} ∽ ∑_{ℙ_{r}^{z}} (△_{i}t) (y_{i−1} (F′ (△_{i}X) A) y_{i}^{−1})^{2} .

By Lemma 6.26 we have

∑_{ℙ_{r}^{z}} (△_{i}t) y_{i−1} (F′′ (△_{i}X) A^{⊗2}) y_{i}^{−1} ∽ ∑_{ℙ_{r}^{z}} (△_{i}t) Ad_{y_{i−1}}F′′ (△_{i}X) A^{⊗2}.

By Lemma 6.27 we have

∑_{ℙ_{r}^{z}} (△_{i}t) (y_{i−1} (F′ (△_{i}X) A) y_{i}^{−1})^{2} ∽ ∑_{ℙ_{r}^{z}} (△_{i}t) Ad_{y_{i−1}} (F′ (△_{i}X) A)^{2} .

Thus the problem reduces to showing

∑_{ℙ_{r}^{z}} (△_{i}t) Ad_{y_{i−1}}F′′ (△_{i}X) A^{⊗2} ∽ ∑_{ℙ_{r}^{z}} (△_{i}t) Ad_{y_{i−1}} (F′ (△_{i}X) A)^{2} .

Invoking the boundedness of the Adjoint operator on (K, ∥·∥_{HS}), and letting J denote the expression

∥∑_{ℙ_{r}^{z}} (△_{i}t) Ad_{y_{i−1}}F′′ (△_{i}X) A^{⊗2} − ∑_{ℙ_{r}^{z}} (△_{i}t) Ad_{y_{i−1}} (F′ (△_{i}X) A)^{2}∥_{HS} ,

we have

J ≤ ∑_{ℙ_{r}^{z}} (△_{i}t) ∥Ad_{y_{i−1}} [F′′ (△_{i}X) A^{⊗2} − (F′ (△_{i}X) A)^{2}]∥_{HS}

≤ C ∑_{ℙ_{r}^{z}} (△_{i}t) ∥F′′ (△_{i}X) A^{⊗2} − (F′ (△_{i}X) A)^{2}∥_{HS} .

However by Lemma 6.28 as r → ∞ we have, P -a.s., that

∑_{ℙ_{r}^{z}} (△_{i}t) ∥F′′ (△_{i}X) A^{⊗2} − (F′ (△_{i}X) A)^{2}∥_{HS} → 0.

Hence we are done.

6.4.4. Proof of Proposition 6.21.

. Let s ∈ [0, z] and let y_{i} (·) denote g_{t_{i}·}^{ℙ_{r}^{z}} as in Definition 6.9. Let F be the matrixexponential map as in Definition 6.9. Let A run through an orthonormal basis of 𝔎.

By Eq. (6.11)

ν_{1}^{ℙ_{r}^{z}}(T, s) ≡ −\frac{1}{1 − z} ∑_{i=1}^{n_{ℙ^{z}_{r}}} Ad_{y_{i−1}(s)}Δ_{i}X(s).

By the definition of H^{2} (z), it will suffice to show that the expression

J_{r} ≡ E [∫_{0}^{·} dσ ∥ν_{1}^{ℙ_{r}^{z}}(T, σ) + \frac{1}{1 − σ} ∫_{0}^{T} Ad_{g_{tσ}}X_{dtσ}∥_{HS}]^{2}

vanishes in the limit. Thus we have:

J_{r} = E [∫_{0}^{·} \frac{dσ}{1 − σ} ∥∫_{0}^{T} Ad_{g_{tσ}}X_{dtσ} − ∑_{i=1}^{n_{ℙ^{z}_{r}}} Ad_{y_{i−1}}Δ_{i}X(σ)∥_{HS}]^{2}

≤ ∫_{0}^{·} \frac{dσ}{(1 − σ)^{2}} ∫_{0}^{·} dσE ∥∫_{0}^{T} Ad_{g_{tσ}}X_{dtσ} − ∑_{i=1}^{n_{ℙ^{z}_{r}}} Ad_{y_{i−1}}Δ_{i}X(σ)∥_{HS}^{2} .

Thus since z < 1, the problem reduces to showing

E ∥∫_{0}^{T} Ad_{g_{tσ}}X_{dtσ} − ∑_{i=1}^{n_{ℙ^{z}_{r}}} Ad_{y_{i−1}}Δ_{i}X(σ)∥_{HS}^{2}

vanishes. Defining

γ_{tσ}^{ℙ_{r}^{z}} ≡ ∑_{i=1}^{n_{ℙ^{z}_{r}}} g_{t_{i−1}σ}^{ℙ_{r}^{z}}1_{(t_{i−1},t_{i}]},

we have only to show that the expression

K_{r} ≡ E ∥∫_{0}^{T} Ad_{g_{tσ}}X_{dtσ} − Ad_{γ}{ℙ^{z}_{r} \atop tσ}X_{dtσ}∥_{HS}^{2}

vanishes in the limit. By Lemma 8.3 X_{dtσ}^{A}X_{dtσ}^{B} = δ_{AB} (σ − σ^{2}) dt. Evaluate our last expression yields

K_{r} = ∑_{p,q} E [∑_{A} ∫_{0}^{T} (Ad_{g_{tσ}}A − Ad_{γ}{ℙ^{z}_{r} \atop tσ}A)_{pq} X_{dtσ}^{A}]^{2}

= ∑_{p,q} ∑_{A} E ∫_{0}^{T} (Ad_{g_{tσ}}A − Ad_{γ}{ℙ^{z}_{r} \atop tσ}A)_{pq}^{2} (σ − σ^{2}) dt

= ∑_{A} (σ − σ^{2}) E ∫_{0}^{T} ∥Ad_{g_{tσ}}A − Ad_{γ}{ℙ^{z}_{r} \atop tσ}A∥_{HS}^{2} dt

≤ C ∑_{A} (σ − σ^{2}) ∫_{0}^{T} E ∥g_{tσ} − γ_{tσ}^{ℙ_{r}^{z}} ∥_{HS}^{2} + E ∥g_{tσ}^{−1} − (γ_{tσ}^{ℙ_{r}^{z}})^{−1}∥_{HS}^{2} dt.

Thus by Dominated convergence and the continuity of inverses, it will suffice to show ∥g_{tσ} − γ_{tσ}^{ℙ_{r}^{z}} ∥_{HS} → 0 P -a.s. Define [t] ≡ t_{i−1} for any t ∈ [t_{i−1}, t_{i}). Then

γ_{tσ}^{ℙ_{r}^{z}} = g_{[t]σ}^{ℙ_{r}^{z}}.

∥g_{tσ} − g_{[t]σ}^{ℙ_{r}^{z}} ∥_{HS} ≤ ∥g_{tσ} − g_{[t]σ}∥_{HS} + ∥g_{[t]σ} − g_{[t]σ}^{ℙ_{r}^{z}} ∥_{HS} .

By Lemma 6.15,

sup_{t∈[0,T]} ∥g^{ℙ_{r}^{z}}(t, σ) − g_{tσ}∥_{HS} → 0, P -a.s.

Thus, for each ω, pick a partition ℙ_{r}^{z} so that ∥g_{·σ} − g_{·σ}^{ℙ_{r}^{z}} ∥_{∞} < ε for all m ≥ n. Pick

m > n so that ∥g_{tσ} − g_{[t]σ}∥_{HS} < ε. Then we see that ∥g_{tσ} − g_{[t]σ}^{ℙ_{r}^{z}} ∥_{HS} → 0 almost surely and so we are done.

6.5. Good partitions(proof of Lemma 6.15).

Theorem 6.29. For any r ∈ ℕ, let t_{i} ≡ \frac{iT}{2^{r}}, and let

ℙ̃_{r} ≡ {0 = t_{0} < ⋯ < t_{2^{r}} = T }

be a partition of [0, T]. Let t ↦ g_{t·} be an L (K)-valued Brownian motion and let gℙ̃_{r} ; ts be the approximation from Definition 6.9. Then gℙ̃_{r}(·, s) converges to g_{·s} in L^{p} for any p ∈ [1, ∞) as r → ∞. Specifically, we have

∥ sup_{t∈[0,T]} ∥gℙ̃_{r}(t, s) − g_{ts}∥_{HS}∥_{L^{p}} → 0 ∀p ∈ [1, ∞) , T < ∞.

This result is a direct consequence of Theorem 7.2 in [21]. See also Wong and Zakai [31].

Proof of Theorem 6.29. View GL_{m} (ℝ) as ℝ^{m^{2}}. Then the path t → X_{ts} is an element of the Wiener space

W_{0} (ℝ^{m^{2}}) ≡ {σ ∈ C ([0, 1] → ℝ^{m^{2}}) |σ (0) = 0} .

Let δ_{r} = T/r. Then Xℙ̃_{r} ; ts is an approximation to the Wiener process t → X_{ts} in the sense of Definition 7.1 of Ikeda and Watanabe [21]. Now apply Theorem 7.2 of [21] to show that as r → ∞,

sup_{t∈[0,T]} ∥gℙ̃_{r} ; ts − g_{ts}∥_{HS} → 0 in L^{2} (Ω) .

Both processes gℙ̃_{r} ; ts, and g_{ts} are K valued and hence bounded in the Hilbert Schmidt norm. Conclude that

sup_{t∈[0,T]} ∥gℙ̃_{r} ; ts − g_{ts}∥_{HS} → 0 in L^{p} (Ω) .

Lemma 6.30. Let {ℙ_{r}} be a sequence of partitions of the interval [0, T] so that |ℙ_{r}| → 0 as r → ∞. Then there exists a subsequence of partitions {ℙ′_{r}} so that

sup_{{ℙ′_{r}}} sup_{1≥s≥σ≥0} |∑_{i=1}^{n_{ℙ′_{r}}} Δ_{i}b^{A}(s)Δ_{i}b^{A}(σ)| < ∞, P -a.s.

Proof of Lemma 6.15. Let \left\{{ \atop ℙ_{r}}\right\} be the sequence of partitions chosen in Theorem

6.29. We will show that there exists a subsequence \{ℙ_{r}^{z}\} of partitions of the \left\{{ \atop ℙ_{r}}\right\} depending only on T and z, with |ℙ_{r}^{z}| ↓ 0 as r → ∞ and

sup_{{ℙ_{r}^{z}}} sup_{σ∈[0,z]} ∑_{i=1}^{n_{ℙ^{z}_{r}}} ∥F (αΔ_{i}X(σ)) − 1∥_{HS}^{2} < ∞ P -a.s.

Since \{ℙ_{r}^{z}\} is a subsequence of the \left\{{ \atop ℙ_{r}}\right\} from Theorem 6.29, we will still have

∥ sup_{t∈[0,T]} ∥g^{ℙ_{r}^{z}}(t, s) − g_{ts}∥_{HS}∥_{L^{p}} → 0 ∀p ∈ [1, ∞) , T < ∞,

as r → ∞ and so we shall be done.

Let \left\{{ \atop ℙ_{r}}\right\} be a refining sequence of partitions, so that |ℙ̃_{r}| ↓ 0 as r → ∞.

∥F (αΔ_{i}X(s)) − 1∥_{HS} ≤ ∑_{j=1}^{∞} \frac{1}{j!} ∥αΔ_{i}X(s)∥_{HS}^{j}

≤ ∥αΔ_{i}X(s)∥_{HS} ∑_{j=0}^{∞} \frac{1}{(j + 1)!} ∥αΔ_{i}X(s)∥_{HS}^{j}

≤ ∥αΔ_{i}X(s)∥_{HS} ∑_{j=0}^{∞} \frac{1}{j!} ∥αΔ_{i}X(s)∥_{HS}^{j} .

By the equivalence of norms in finite dimensions there is some finite constant so that for any A ∈ 𝔎, we have ∥A∥_{HS} ≤ C ∥A∥_{𝔎}. Thus

∥F (αΔ_{i}X(s)) − 1∥_{HS} ≤ ∥αΔ_{i}X(s)∥_{HS} exp {C |α| ∥Δ_{i}X(s)∥_{𝔎}} . (6.20)

But picking a particular orthonormal basis {A} of 𝔎, we have

∥Δ_{i}X(s)∥_{𝔎}^{2} = ∑_{A} (Δ_{i}X^{A}(s))^{2} .

ON THE EQUIVALENCE OF MEASURES ON LOOP SPACE 73

By Eq. [3.5] we see that

∥Δ_{i}X(s)∥_{𝔎}^{2} = ∑_{A} [Δ_{i}b^{A}(s) − ∫_{0}^{s} Δ_{i}b^{A}(σ)\frac{(1 − s)}{(1 − σ)^{2}} dσ]^{2}
≤ ∑_{A} [|Δ_{i}b^{A}(s)| + ∫_{0}^{s} |Δ_{i}b^{A}(σ)| \frac{(1 − s)}{(1 − σ)^{2}} dσ]^{2}
≤ ∑_{A} [ sup_{(τ,σ)∈R_{T s}} |b_{τσ}^{A}| (1 + ∫_{0}^{s} \frac{(1 − s)}{(1 − σ)^{2}} dσ)]^{2} ,

where we have let R_{T s} denote the rectangle [0, T] × [0, s] in ℝ^{2}. Now

∫_{0}^{s} \frac{(1 − s)}{(1 − σ)^{2}} dσ = (1 − s) ∫ d (\frac{1}{1 − σ} ) = s ≤ 1,

and therefore

∥Δ_{i}X(s)∥_{𝔎}^{2} ≤ 4 ∑_{A} sup_{(τ,σ)∈R_{1,1}} |b_{τσ}^{A}|^{2} . (6.21)

By Theorem 6.4 we have

E sup_{R_{1,1}} |b_{τσ}^{A}|^{2} ≤ 2^{4} sup_{R_{1,1}} E |b_{τσ}^{A}|^{2} ≤ 2^{4} < ∞. (6.22)

Therefore by Eq. [6.21]

exp [C |α| ∥Δ_{i}X(s)∥_{𝔎}] ≤ exp [C ∑_{A} sup_{R_{1,1}} |b_{τσ}^{A}|^{2}] =: C̃.

By Eq. [6.22],

C̃ < ∞, P -a.s.

Thus returning to Eq. [6.20], we have P -a.s..,

∑_{i=1}^{nℙ̃r} ∥F (αΔ_{i}X(s)) − 1∥_{HS}^{2} ≤ ∥αΔ_{i}X(s)∥_{HS}^{2} exp C |α| ∥Δ_{i}X(s)∥_{𝔎}
≤ C̃ ∑_{i=1}^{nℙ̃r} ∥αΔ_{i}X(s)∥_{HS}^{2}
(6.23)  ≤ C̃ ∑_{A} ∑_{i=1}^{nℙ̃r} |Δ_{i}X^{A}(s)|^{2} .

C̃ is independent of the partition sequence \left\{{ \atop ℙ_{r}}\right\} as well as the partition points {t_{i}}. Thus it will suffice to show that we can find a subsequence of partitions {ℙ^{z}} so that

sup_{r} sup_{[0,1−ε]} ∑_{A} ∑_{i=1}^{n_{ℙ^{z}_{r}}} |Δ_{i}X^{A}(s)|^{2} < ∞. (6.24)

Let \{ℙ_{r}^{1}\} be the subsequence of \left\{{ \atop ℙ_{r}}\right\} from Lemma 6.30. By Ito’s Lemma,

∑_{i=1}^{n_{ℙ^{1}_{r}}} |Δ_{i}X^{A}(s)|^{2} = sT + ∑_{i=1}^{n_{ℙ^{1}_{r}}} ∫_{0}^{s} 2Δ_{i}X^{A}(σ)Δ_{i}X^{A}(dσ). (6.25)

By Theorem 3.19

Δ_{i}X^{A} (σ) = Δ_{i}b^{A} (σ) − ∫_{0}^{σ} Δ_{i}b^{A} (u) \frac{(1 − σ)}{(1 − u)^{2}} du

and so

Δ_{i}X^{A} (dσ) = Δ_{i}b^{A} (dσ) − \frac{Δ_{i}b^{A} (σ)}{(1 − σ)^{2}}dσ + ∫_{0}^{σ} \frac{Δ_{i}b^{A} (u) du}{(1 − u)^{2}}dσ.

Therefore

∫_{0}^{s} ∑_{i=1}^{n_{ℙ^{1}_{r}}}Δ_{i}X^{A}(σ)Δ_{i}X^{A}(dσ)

≤ ∫_{0}^{s} ∑_{i=1}^{n_{ℙ^{1}_{r}}} Δ_{i}X^{A}(σ)Δ_{i}b^{A} (dσ) − ∫_{0}^{s} ∑_{i=1}^{n_{ℙ^{1}_{r}}} \frac{Δ_{i}X^{A}(σ)Δ_{i}b^{A} (σ)}{(1 − σ)^{2}}dσ

+ ∫_{0}^{s} ∫_{0}^{σ} ∑_{i=1}^{n_{ℙ^{1}_{r}}} \frac{Δ_{i}X^{A}(σ)Δ_{i}b^{A} (u) du}{(1 − u)^{2}}dσ

= I_{1} − I_{2} + I_{3}. (6.26)

I_{1} = ∫_{0}^{s} ∑_{i=1}^{n_{ℙ^{1}_{r}}} Δ_{i}X^{A}(σ)Δ_{i}b^{A} (dσ) is an 𝔉_{1s}-martingale.

By Doob’s L^{p}-inequality, we have

E [ sup_{s∈[0,1]} I_{1}]^{2} ≤ 2E [∫_{0}^{1} ∑_{i=1}^{n_{ℙ^{1}_{r}}} Δ_{i}X^{A}(σ)Δ_{i}b^{A} (dσ)]^{2}
= 2 ∑_{i=1}^{n_{ℙ^{1}_{r}}} (Δ_{i}t)^{2} ∫_{0}^{1} G_{0} (σ, σ) dσ
→ 0 as r → ∞.

Thus we can find a subsequence {ℙ_{r}^{2}} of {ℙ_{r}^{1}} so that sup_{s∈[0,1]} |I_{1}| → 0 P -a.s. as r → ∞ and so that

sup_{r} sup_{s∈[0,1]} |I_{1}| < ∞ P -a.s.

ON THE EQUIVALENCE OF MEASURES ON LOOP SPACE 75

Applying Theorem 3.19 to I_{2} yields

sup_{r} sup_{s∈[0,1−ε]} |I_{2}|

= sup_{r} sup_{s∈[0,1−ε]} |∫_{0}^{s} ∑_{i=1}^{n_{ℙ^{2}_{r}}} \frac{Δ_{i}X^{A}(σ)Δ_{i}b^{A} (σ)}{(1 − σ)^{2}}dσ|

≤ sup_{r} sup_{s∈[0,1]} |∑_{i=1}^{n_{ℙ^{2}_{r}}} Δ_{i}b^{A} (σ) Δ_{i}b^{A} (σ)| ∫_{0}^{1−ε} \frac{dσ}{(1 − σ)^{2}}

+ sup_{r} sup_{s∈[0,1]} |∑_{i=1}^{n_{ℙ^{2}_{r}}} Δ_{i}b^{A} (σ) Δ_{i}b^{A} (u)| ∫_{0}^{1−ε} ∫_{0}^{σ} \frac{dσdu}{(1 − σ)(1 − u)^{2}} .

This expression is finite by Lemma 6.30. A last invocation of Theorem 3.19 and Lemma 6.30 gives

sup_{r} sup_{s∈[0,1−ε]} |I_{3}|

= sup_{r} sup_{s∈[0,1−ε]} |∫_{0}^{s} ∫_{0}^{σ} ∑_{i=1}^{n_{ℙ^{2}_{r}}} \frac{Δ_{i}X^{A}(σ)Δ_{i}b^{A} (u) du}{(1 − u)^{2}}dσ|

≤ sup_{r} sup_{s∈[0,1]} |∑_{i=1}^{n_{ℙ^{2}_{r}}} Δ_{i}b^{A} (σ) Δ_{i}b^{A} (u)| ∫_{0}^{1−ε} ∫_{0}^{σ} \frac{dudσ}{(1 − u)^{2}}

+ sup_{r} sup_{s∈[0,1−ε]} |∑_{i=1}^{n_{ℙ^{2}_{r}}} Δ_{i}b^{A} (u) Δ_{i}b^{A} (ν)|

× ∫_{0}^{1−ε} ∫_{0}^{σ} ∫_{0}^{σ} \frac{(1 − σ)dνdudσ}{(1 − ν)^{2}(1 − u)^{2}}

< ∞.

Therefore returning to 6.26 we see that

sup_{r} sup_{s∈[0,1−ε]} |∫_{0}^{s} ∑_{i=1}^{n_{ℙ^{2}_{r}}} Δ_{i}X^{A}(σ)Δ_{i}X^{A}(dσ)| < ∞ P -a.s.

Thus by Eq. [6.25] we see that

sup_{r} sup_{s∈[0,1−ε]} ∑_{i=1}^{n_{ℙ^{2}_{r}}} |Δ_{i}X^{A}(s)|^{2} < ∞ P -a.s.

Taking ℙ_{r}^{z} = ℙ_{r}^{2} we see that Eq. [6.24] is satisfied and so we are done. Proof of Lemma 6.30. By Ito’s Lemma,

∑_{i=1}^{n_{ℙr}} |Δ_{i}b^{A}(s)|^{2} = sT + ∫_{0}^{s} ∑_{i=1}^{n_{ℙr}} 2Δ_{i}b^{A}(σ)Δ_{i}b^{A}(dσ). (6.27)

76 VIKRAM K. SRIMURTHY

Therefore,

sup_{[0,1]} ∑_{i=1}^{n_{ℙr}} |Δ_{i}b^{A}(s)|^{2} = 1 + sup_{[0,1]} |∫_{0}^{s} ∑_{i=1}^{n_{ℙr}} 2Δ_{i}b^{A}(σ)Δ_{i}b^{A}(dσ)| .

Since the process

s → ∫_{0}^{s} ∑_{i=1}^{n_{ℙr}} 2Δ_{i}b^{A}(σ)Δ_{i}b^{A}(dσ)

is an 𝔉_{1s}-martingale, by Doob’s L^{p}-inequality we have

E sup_{s∈[0,1]} |∫_{0}^{s} ∑_{i=1}^{n_{ℙr}} Δ_{i}b^{A}(σ)Δ_{i}b^{A}(dσ)|^{2}

≤ 4E |∫_{0}^{1} ∑_{i=1}^{n_{ℙr}} Δ_{i}b^{A}(σ)Δ_{i}b^{A}(dσ)|^{2}

≤ ∑_{i=1}^{n_{ℙr}} ∫_{0}^{1} E (Δ_{i}b^{A})^{2} (σ)Δ_{i}tdσ

= ∑_{i=1}^{n_{ℙr}} ∫_{0}^{1} (Δ_{i}t)^{2} σdσ ≤ \frac{1}{2} |ℙ_{r}| → 0.

Therefore there exists a subsequence of partitions {ℙ_{r}^{1}} so that the expression

sup_{s∈[0,1]} |∫_{0}^{s} ∑_{i=1}^{n_{ℙ^{1}_{r}}} Δ_{i}b^{A}(σ)Δ_{i}b^{A}(dσ)| → 0, P -a.s. as r → ∞.

Returning to Eq.[6.27]we see that

(6.28)  sup_{r} sup_{s∈[0,1]} ∑_{i=1}^{n_{ℙ^{1}_{r}}} |Δ_{i}b^{A}(s)|^{2} < ∞, P -a.s.

If s ≥ σ then

s → ∑_{i=1}^{n_{ℙ^{1}_{r}}} [Δ_{i}b^{A}(s) − Δ_{i}b^{A}(σ)] Δ_{i}b^{A}(σ) is an 𝔉_{1s}-martingale,

and so by Doob’s inequality, we have

E sup_{s∈[0,1]} |∑_{i=1}^{n_{ℙ^{1}_{r}}} [Δ_{i}b^{A}(s) − Δ_{i}b^{A}(σ)] Δ_{i}b^{A}(σ)|^{2}

≤ 2E |∑_{i=1}^{n_{ℙ^{1}_{r}}} [Δ_{i}b^{A}(1) − Δ_{i}b^{A}(σ)] Δ_{i}b^{A}(σ)|^{2} .

By incremental independence of the Brownian sheet, this previous expression is just

= E ∑_{i=1}^{n_{ℙ^{1}_{r}}} E [Δ_{i}b^{A}(1) − Δ_{i}b^{A}(σ)]^{2} E (Δ_{i}b^{A})^{2} (σ)

= E ∑_{i=1}^{n_{ℙ^{1}_{r}}} (Δ_{i}t)^{2} (1 − σ)σ
→ 0 as r → ∞.

Thus there is a subsequence {ℙ_{r}^{2}} so that

(6.29)  sup_{r} sup_{s>σ} ∑_{i=1}^{n_{ℙ^{2}_{r}}} [Δ_{i}b^{A}(s) − Δ_{i}b^{A}(σ)] Δ_{i}b^{A}(σ) < ∞ P -a.s.

If 1 ≥ s ≥ σ, we can always replace a given sequence {ℙ}of partitions by a subsequence {ℙ_{r}^{2}} so that

sup_{r} sup_{1≥s≥σ≥0} |∑_{i=1}^{n_{ℙ^{2}_{r}}} Δ_{i}b^{A}(s)Δ_{i}b^{A}(σ)|

≤ sup_{r} sup_{1≥s≥σ} |∑_{i=1}^{n_{ℙ^{2}_{r}}} [Δ_{i}b^{A}(s) − Δ_{i}b^{A}(σ)] Δ_{i}b^{A}(σ)| + sup_{r} sup_{σ∈[0,1]} ∑_{i=1}^{n_{ℙ^{2}_{r}}} (Δ_{i}b^{A})^{2} (σ)

< ∞ P -a.s. by Eqs. [6.29] and [6.28].

Letting {ℙ′} denote this subsequence {ℙ_{r}^{2}}, we are done.

7. HKM ↓_{𝔉_{s}}∽ P W M ↓_{𝔉_{s}}

Theorem 7.1. ν_{1} (e, ·)[Heat Kernel measure on L (K)] is equivalent to μ_{0}[Pinned Wiener Measure on L (K)] as measures on (L (K) , 𝔊_{z}) where 𝔊_{z} ≡ σ ⟨x_{t} : t ∈ [0, z]⟩, for any z < 1.

We supply the proof of this result at the end of this section.

Definition 7.2. Let B_{ts} be defined to solve the Fisk-Stratonowicz equation B_{tδs} = b_{tδs}B_{ts} with B_{t0} = e where b is the Brownian sheet from Theorem 3.19. By the following Remark, we see that t → B_{ts} is a Brownian motion on K with parameter s.

Remark 7.3 (t → B_{ts} is a Brownian motion on K). Let \overline{h}_{ts} solve \overline{h}_{tδs} = b_{tδs}\overline{h}_{ts} with \overline{h}_{t0} = e. Let h̃_{ts} ≡ \overline{h}_{st} and b̃_{ts} ≡ b_{st}. Then s → \overline{h}_{ts} is the same process as s → h̃_{st} and so \overline{h}_{tδs} = h̃_{δst}. Similarly, b_{tδs} = b̃_{δst}. Thus h̃_{st} solves h̃_{δst} = b̃_{δst}h̃_{st} with h̃_{0t} = e. To put it another way, h̃_{ts} solves h̃_{δts} = b̃_{δts}h̃_{ts} with h̃_{0s} = e. Then h ≡ h̃^{−1} solves h_{δts} = −h_{ts}b̃_{δts} with h_{0s} = e. By Lemma 3.9 if β is a 𝔎-valued Brownian sheet and h_{ts} solves h_{δts} = h_{ts}β_{δts} with h_{0s} = e then the process s ↦ h_{ts} is a K-valued Brownian motion with parameter t. Taking β = −b̃, we see that s ↦ h̃_{ts}^{−1} is a K-valued Brownian motion with parameter t. Thus s ↦ h̃_{ts} is also a K-valued Brownian motion with parameter t and so s ↦ \overline{h}_{st} is a Brownian motion on K with parameter t. Switching t and s yields t ↦ \overline{h}_{ts} is a Brownian motion on K with parameter s.

Remark 7.4. Let π_{s} : C ([0, 1] → L) → C ([0, s] → L) ; π_{s} (x_{·}) (r) = x (r) for any r ≤ s. We make no distinction between a measure ν_{1} on (C ([0, s] → L) , σ ⟨x_{r} : r ≤ s⟩) and a measure ν_{2} on (C ([0, 1] → L) , σ ⟨x_{r} : r ≤ s⟩) so long as ν_{1} (F ∘ π_{s}) = ν_{2} (F) for any F : C ([0, s] → L) → ℝ. where L stands for either K or 𝔎.

Lemma 7.5. If κ_{1} ∽ κ_{2} then κ_{1} ⊗ ν ∽ κ_{2} ⊗ ν, where κ_{1}, κ, ν are probability measures.

Proof. Will suffice to show that if κ_{1} ≪ κ_{2} then κ_{1}⊗ν ≪ κ_{2}⊗ν. For rectangles, it is clear that (κ_{1} ⊗ ν) (1_{A} (x) 1_{B} (y)) = (κ_{2} ⊗ ν) (1_{A} (x) f (x) 1_{B} (y)). This extends to linear combinations of rectangles by linearity and all bounded measurable functions by dominated convergence. Thus d (κ_{1} ⊗ ν) /d (κ_{2} ⊗ ν) (x, y) = dκ_{1}/d (κ_{2}) (x). Thus κ_{1} ⊗ ν ≪ κ_{2} ⊗ ν.

Theorem 7.6. Let t ↦ g_{t·} be our L (K)-valued Brownian motion from Definition 2.22 and let s → B_{ts} be the K-valued Brownian motion of Definition 7.2. Then g_{T ·} and B_{T ·} have equivalent laws as measures on C ([0, s] → K) for any s < \frac{\sqrt{1+4T} −1}{2T}.

We prove this result after the proof of Theorem 7.7.

Theorem 7.7. Law Y_{T ·} ∽ Law b_{T ·} as measures on C ([0, s] → 𝔎) for any s < \frac{\sqrt{1+4T} −1}{2T}. Here the random variable

Y_{T ·} ≡ ∫_{R_{T ·}} Ad_{g_{tσ}}b_{dtdσ} − ∫_{0}^{·} \frac{dσ}{1 − σ} ∫_{0}^{T} Ad_{g_{tσ}}X_{dtσ}

is as in Theorem 6.11.

The proof of Theorem 7.7 is given after that of Lemma 7.11.

Remark 7.8. Since for s < 1, Law X_{T.} ∽ Law b_{T.} (as measures on C ([0, s] → 𝔎)), one might suspect Law X ∽ Law b (as measures on C ([0, 1] × [0, s] → 𝔎)) which should then indicate that

Law Y_{T} . = Law ∫_{R_{T ·}} Ad_{g_{tσ}}X_{dtdσ} ∽ Law ∫_{R_{T ·}} Ad_{g_{tσ}}b_{dtdσ} = Law (b_{T ·}) .

Unfortunately in the t-variable, X_{.s} and b_{.s} are Brownian motions with parameters s − s^{2} and s respectively.

Thus LawX ⊥ Lawb since

P_{X} (∑_{i} |Δ_{i}ω(s)|^{2} → s − s^{2}) = 1,

while

P_{b} (∑_{i} |Δ_{i}ω(s)|^{2} → s) = 1.

Hence these two measures live on different sets.

The proof of Theorem 7.7 relies heavily on Girsanov’s Theorem which we state here for convenience.

Theorem 7.9 (Girsanov, see [22]). Let (Ω, 𝔉, {𝔉_{·}} , P) be a filtered probability space. Let β_{·} be a d-dimensional Brownian motion and let Z_{·} be an ℝ^{d}-valued adapted process so that E exp \frac{1}{2} ∫_{0}^{S} |Z_{s}|^{2} ds is finite and ∫_{0}^{S} (Z_{s}^{i})^{2} ds < ∞ almost surely for any i ∈ {1, ⋯, d}. Define

Z_{·} ≡ exp [∫_{0}^{·} Z_{s} · dβ_{s} − \frac{1}{2} ∫_{0}^{·} |Z_{s}|^{2} ds] .

Define a new measure P̃_{S} on 𝔉_{S} by setting P̃ (A) = E1_{A}Z_{S}. Then P̃_{S} is a probability equivalent to P and the process {Y_{t}, 𝔉_{t}; 0 ≤ s ≤ S} is a d-dimensional Brownian motion on (Ω, 𝔉_{S}, P̃) where Y_{·} ≡ β_{·} + ∫_{0}^{·} Z_{s}ds.

Lemma 7.10. The expression X̃_{tσ} ≡ ∫_{0}^{t} Ad_{g_{uσ}} X_{duσ} has the same law as X_{tσ}.

Proof. X̃_{tσ} is a 𝔉_{tσ} martingale. To show X_{·s} and X̃_{·s} have the same law it will suffice to show X̃_{·σ} is a 𝔎-valued Brownian motion with parameter σ − σ^{2}. To this end, let {A} run through an orthonormal basis of 𝔎. Then

X̃_{dtσ} ⊗ X̃_{dtσ} = Ad_{g_{tσ}} X_{dtσ} ⊗ Ad_{g_{tσ}} X_{dtσ}
= (σ − σ^{2}) dt ∑_{A,B} δ_{AB}Ad_{g_{tσ}} A ⊗ Ad_{g_{tσ}} B
= (σ − σ^{2}) dt ∑_{A} (Ad_{g_{tσ}} A)^{⊗2}
= (σ − σ^{2}) dt ∑_{A} A^{⊗2}.

Thus we are done.

Lemma 7.11. Let X be the 𝔎-valued Brownian bridge sheet of Theorem 2.25. Then

E exp [\frac{1}{2} ∫_{0}^{s} dσ |∫_{0}^{T} Ad_{g_{tσ}} \frac{X_{dtσ}}{1 − σ} |_{𝔎}^{2}] < ∞, if s < \frac{\sqrt{1 + 4T} − 1}{2T}.

80 VIKRAM K. SRIMURTHY

Proof.

E exp [\frac{1}{2} ∫_{0}^{s} dσ |∫_{0}^{T} Ad_{g_{tσ}} \frac{X_{dtσ}}{1 − σ} |_{𝔎}^{2}]

= E ∑_{P ≥0} [\frac{1}{2} ∫_{0}^{s} dσ |∫_{0}^{T} Ad_{g_{tσ}} \frac{X_{dtσ}}{1 − σ} |_{𝔎}^{2}]^{p}

≤ ∑_{P ≥0} \frac{s^{p−1}}{p!2^{p}} E ∫_{0}^{s} dσ |∫_{0}^{T} Ad_{g_{tσ}} \frac{X_{dtσ}}{1 − σ} |_{𝔎}^{2p} by Hölder’s Inequality

= \frac{1}{s} ∑_{P ≥0} \frac{s^{p}}{p!2^{p}} ∫_{0}^{s} dσE |∫_{0}^{T} Ad_{g_{tσ}} \frac{X_{dtσ}}{1 − σ} |_{𝔎}^{2p}

= \frac{1}{s} ∑_{P ≥0} \frac{s^{p}}{p!2^{p}} ∫_{0}^{s} dσE |∫_{0}^{T} \frac{X_{dtσ}}{1 − σ} |_{𝔎}^{2p}

= \frac{1}{s} ∫_{0}^{s} dσ ∑_{P ≥0} \frac{1}{p!}E (\frac{s |X_{T σ}|_{𝔎}^{2}}{2 (1 − σ)^{2}} )^{p}

= \frac{1}{s} ∫_{0}^{s} dσE exp (\frac{s |X_{T σ}|_{𝔎}^{2}}{2 (1 − σ)^{2}} )

= \frac{1}{s} ∫_{0}^{s} dσ ∫_{ℝ^{dim 𝔎}} exp (\frac{s |x|^{2}}{2 (1 − σ)^{2}} ) exp (\frac{− |x|^{2}}{2T σ (1 − σ)}) \frac{dx}{[2πT σ (1 − σ)]^{\frac{dim 𝔎}{2}}}

= \frac{1}{s} ∫_{0}^{s} dσ ∫_{ℝ^{dim 𝔎}} exp [−\frac{|x|^{2}}{2T σ (1 − σ)} (1 − \frac{sT σ}{(1 − σ)})] \frac{dx}{[2πT σ (1 − σ)]^{\frac{dim 𝔎}{2}}}

= \frac{1}{s} ∫_{0}^{s} dσ [2πT σ (1 − σ)]^{− \frac{dim 𝔎}{2}}

< ∞ ⟺ 1 − \frac{sT σ}{(1 − σ)} > 0, ∀σ ∈ [0, s] .

1−\frac{sT σ}{(1 − σ)} > 0 for σ ∈ [0, s]

⟺ \frac{σ}{1 − σ} < \frac{1}{sT} for σ ∈ [0, s]

⟺ \frac{s}{1 − s} < \frac{1}{sT}

⟺ T s^{2} + s − 1 < 0

⟺ s ∈ [0, \frac{\sqrt{1 + 4T} − 1}{2T}) .

We are now able to prove Theorem 7.7.

Proof. Define

β_{T s} = ∫_{R_{T s}} Ad_{g_{tσ}}b_{dtdσ};

Z_{T} (σ) ≡ \frac{−1}{(1 − σ)} ∫_{0}^{T} Ad_{g_{tσ}} X_{dtσ}.

By definition of Y_{T ·} in Theorem 6.11

Y_{T ·} ≡ β_{T ·} + ∫_{0}^{·} dσZ_{T} (σ) .

By Lemma 7.11,

E exp ∫_{0}^{S} |Z_{T} (σ)|_{𝔎}^{2} dσ < ∞ whenever S < \frac{\sqrt{1 + 4T} − 1}{2T}.

Thus the measure

dP̃_{S} ≡ exp [∫_{0}^{S} Z_{T} (s) · dβ_{T s} − \frac{1}{2} ∫_{0}^{S} |Z_{T} (s)|^{2} ds] dP

is a probability on 𝔉_{T S} and the process {Y_{T s}, 𝔉_{T s}; 0 ≤ s ≤ S} is a P̃_{S}-Brownian motion on 𝔎. Thus for any set A ⊂ (C [0, S] → 𝔎)

E1_{A} ∘ β_{T ·} = 0 ⟺ Ẽ1_{A} ∘ Y_{T ·} = 0 ⟺ E1_{A} ∘ Y_{T ·} = 0,

since the measures P̃_{S} and P are equivalent on 𝔉_{T S}. [Note:- it is essential that A only depend on the path to time S or else 1_{A} ∘ Y_{T ·} will cease to be 𝔉_{T S}- measurable.]

We now return to the proof of Theorem 7.6.

Proof. Fix s. Pick T so that s < \frac{\sqrt{1+4T} −1}{2T}.

Define a map Γ from C ([0, z] → 𝔎) to C ([0, z] → K) so that Γ (x_{·}) = y_{·}, where y_{δs} = x_{δs}y_{s} with y_{0} = e, the integration being done with respect to the Wiener Measure on C ([0, z] → 𝔎) with parameter T.

If we can show that Γ : b_{T ·} ↦ B_{T ·}, and Γ : Y_{T ·} ↦ g_{T ·} we shall be done. This is so because by Theorem 7.7 Law Y_{T ·} is equivalent to Law b_{T ·}. Thus

E1_{A} ∘ g_{T ·} = 0
⟺ E1_{Γ_{1}^{−1}(A)} ∘ Y_{T ·} = 0
⟺ E1_{Γ_{1}^{−1}(A)} ∘ b_{T ·} = 0
⟺ E1_{A} ∘ B_{T ·} = 0.

Hence by the Radon-Nikodym Theorem Law g_{T ·} is equivalent to Law B_{T ·}.

To show Γ : b_{T ·} ↦ B_{T ·}, and Γ : Y_{T ·} ↦ g_{T ·}, we shall invoke Lemma 8.1. Let Ω_{0} be C ([0, 1] → 𝔎) where K is identified with ℝ^{dim 𝔎}. Let (Ω, 𝔉, {𝔉_{T ·}} , P) be our standard probability space as in Definition 2.22. The stochastic differential equation we will use in Lemma 8.1 will be

y_{ds} = x_{δs}y_{s} = x_{ds}y_{s} + \frac{T ds}{2} (∑_{A} A^{2}) y_{s} with y_{0} = 1.

Here the {A} run through an orthonormal basis of the Lie algebra 𝔎. Clearly the boundedness conditions of the Lemma are satisfied. Also both Law b_{T ·} and Law Y_{T ·} are absolutely continuous with respect to μ′ = Law b_{T ·}. Thus either b_{T ·} or Y_{T ·} can be taken to be the X_{·} in the Lemma. Thus we see that Γ (b_{T ·}) (δs) = b_{T δs}Γ (b_{T ·}) (δs) and Γ (Y_{T ·}) (δs) = Y_{T δs}Γ (Y_{T ·}) (δs). By Definition 7.2 B_{tδs} ≡ b_{tδs}B_{ts} with B_{t0} = 1. Hence Γ (b_{T ·}) = B_{T ·}. By Theorem 6.11, g_{tδs} ≡ Y_{tδs}g_{ts} with g_{t0} = 1. Hence Γ (Y_{T ·}) = g_{T ·} and so we are done.

We now return to the proof of Theorem 7.1.

Proof. Let μ_{0} be Pinned Wiener measure on L (K) and let μ be Wiener measure on C ([0, 1] → K) as in Definitions 2.7 and 2.9. Then μ = Law [B_{1·}] since B_{1·} is a standard K-valued Brownian motion by Definition 7.2. A key fact that we shall exploit in this proof is μ_{0} is equivalent to μ on 𝔊_{z} for any z < 1.

Fix z < 1.Now lim_{T →0} \frac{\sqrt{1+4T} −1}{2T} = 1 so there exists an N ∈ ℕ large so that

z < ^{\frac{\sqrt{1+4/N}−1}{2/N}}. Let T ≡ 1/N. Let F : C ([0, 1] → K) → ℝ so that F ∈ 𝔊_{z}. Then

ν_{1} (e, A) = P {g_{1·} ∈ A}
= P {g_{(1/N)·} (g_{(1/N)·}^{−1}g_{(2/N)·}) ⋯ (g_{(N−1/N)·}^{−1}g_{1·}) ∈ A}
= (⊗_{i=1}^{N}Law_{g_{T ·}}) (k_{1} ⋯ k_{N} ∈ A) ,

where A′ ≡ {(k_{1}, ⋯, k_{n}) : k_{1} ⋯ k_{N} ∈ A}. Now by Theorem 7.6, since the condition z < \frac{\sqrt{1+4T} −1}{2T} obtains, g_{T ·} has a law equivalent to that of B_{T ·}, on the restricted σ-algebra 𝔊_{z}. Invoking Lemma 7.5 repeatedly, we see that ⊗_{i=1}^{N}Law_{g_{T ·}} ∽ ⊗_{i=1}^{N}Law_{B_{T ·}}, on the restricted σ-algebra 𝔊_{z}. Thus if A is 𝔊_{z}-measurable, ν_{1} (e, A) = 0 holds if and only if (⊗_{i=1}^{N}Law_{B_{T ·}}) (k_{1} ⋯ k_{N} ∈ A) = 0 holds. Since t ↦ B_{ts} is a K-valued Brownian motion (see Remark 7.3), it exhibits incremental independence. Thus (Law_{B_{1·}}) (A) = 0 if and only if (⊗_{i=1}^{N}Law_{B_{T ·}}) (k_{1} ⋯ k_{N} ∈ A) = 0. Thus we have ν_{1} (e, ·) ↓ 𝔊_{z} ∽ μ ↓ 𝔊_{z} ∽ μ_{0} ↓ 𝔊_{z} and so we are done.

8. Appendix

8.1. General Technical results.

Lemma 8.1 (General Technical Lemma). Let X be an (Ω, 𝔉_{t}, P) continuous semimartingale taking values in ℝ^{d}. Ω_{0} ≡ C ([0, 1] → ℝ^{d}) is to be thought of as the measure space (Ω_{0}, ℌ_{t}, μ′) where Law X_{·} ≪ μ′, and ℌ_{t} ≡ σ ⟨X_{r} : r ≤ t⟩. Let x, ω denote members of the probability spaces Ω_{0} and Ω respectively. Let a be an ℝ^{d}- valued Ω_{0}-random variable that solves the following stochastic differential equation

a (dt, x) = ∑_{j} c (t, a (t) , x) r (dt, x) + c^{0} (t, a (t) , x) dt with a_{i} (0, x) = K_{i}, (8.1)

where c ∈ C_{b} (ℝ × ℝ^{d} × Ω_{0} → ℝ^{d} ⊗ ℝ^{d∗}) , c^{0} ∈ C_{b} (ℝ × ℝ^{d} × Ω_{0} → ℝ^{d}). Then A (t, ω) ≡ a (t, X_{·} (ω)) solves the Stochastic differential equation.

A (dt, ω) = ∑_{j} C (t, A (t, ω) , ω) R (dt, ω) + C^{0} (t, A (t, ω) , ω) dt (8.2)

with A_{i} (0, ω) = K_{i}; such that C (t, ξ, ω) ≡ c (t, ξ, X_{·} (ω)), R (t, ω) ≡ r (t, X_{·} (ω)), C^{0} (t, ξ, ω) ≡ c^{0} (t, ξ, X_{·} (ω)), and ξ ∈ ℝ^{d}.

Proof. For convenience, let μ ≡ Law X_{·}, For explicitness we will not suppress dependence on the probability space as is done traditionally.

a_{i} (T, x) = K_{i} + ∫_{0}^{T} ∑_{j} c_{ij} (t, x, a (t)) r_{i} (dt, x) + c_{i}^{0} (t, x, a (t)) dt.

This means that the expression

∑_{k} ∑_{j} c_{ij} (t_{k−1}^{ℙ}, x, a (t_{k−1}^{ℙ})) (r_{i} (t_{k}^{ℙ}, x) − r_{i} (t_{k−1}^{ℙ}, x))

converges in L^{2} (μ′) as |ℙ| → 0 to the expression

a_{i} (T, x) − K_{i} − ∫_{0}^{T} c_{i}^{0} (t, x, a (t)) dt.

Since L^{2}-convergence implies convergence in measure, we see that the expression

∑_{k} ∑_{j} c_{ij} (t_{k−1}^{ℙ}, x, a (t_{k−1}^{ℙ})) (r_{i} (t_{k}^{ℙ}, x) − r_{i} (t_{k−1}^{ℙ}, x))

converges in the measure μ′ as |ℙ| → 0 to

a_{i} (T, x) − K_{i} − ∫_{0}^{T} c_{i}^{0} (t, x, a (t)) dt.

Since μ ≪ μ′, the following statement holds. For any ε > 0, there exists a δ (ε) so that if μ′ (A) < δ (ε) then μ (A) < ε.(if this were not so can have μ′ (A) = 0 and μ (A) > ε for some ε). Hence

lim_{|ℙ|→0}μ (| \begin{matrix}∑_{k} ∑_{j} c_{ij} (t_{k−1}^{ℙ}, x, a (t_{k−1}^{ℙ})) (r_{i} (t_{k}^{ℙ}, x) − r_{i} (t_{k−1}^{ℙ}, x)) \\ − (a_{i} (T, x) − K_{i} − ∫_{0}^{T} c_{i}^{0} (t, x, a (t)) dt) \end{matrix} | < ε) = 0.

This in turn implies that as |ℙ| → 0,

P (| \begin{matrix}∑_{kj} C_{ij} (t_{k−1}^{ℙ}, ω, a (t_{k−1}^{ℙ}, X (·, ω))) (R_{i} (t_{k}^{ℙ}, ω) − R_{i} (t_{k−1}^{ℙ}, ω)) \\ +K_{i} + ∫_{0}^{T} c_{i}^{0} (t, X (·, ω) , a (t, X (·, ω))) dt − a_{i} (T, X (·, ω)) \end{matrix} | < ε) → 0.

However since, essentially by assumption, the expression

K_{i} + ∫_{0}^{T} C_{i}^{0} (t, X (·, ω) , a (t, X (·, ω))) dt

+ ∑_{kj} C_{ij} (t_{k−1}^{ℙ}, ω, a (t_{k−1}^{ℙ}, X (·, ω))) (R_{i} (t_{k}^{ℙ}, ω) − R_{i} (t_{k−1}^{ℙ}, ω))

has an L^{2}-limit, it must be equal to a_{i} (T, X (·, ω)). Hence A_{i} (T, ω) ≡ a_{i} (T, X (·, ω)) satisfies Eq. (8.2) as desired.

Theorem 8.2 (Kolmogorov’s extension Theorem). Let (𝔼, d) be a complete metric space and let U_{x} be an 𝔼-valued process for all dyadic rationals x in ℝ^{n}. Suppose that for all x, y we d (U_{x}, U_{y}) is a random variable and that there exist strictly positive constants ε, c, β so that

P [d (U_{x}, U_{y})^{ε}] ≤ C ∥x − y∥^{n+β} .

Then P -a.s. the function x → U_{x} can be extended uniquely to a continuous function from ℝ^{n} to 𝔼.

Proof. See Theorem 53 of Chapter IV of Protter [30].

8.2. Brownian Sheets and bridges.

Lemma 8.3 (Quadratic Variations). We compute some quadratic variations we shall later find useful. Let A, B be two perpendicular unit vectors. As in Definitions 3.5 and 2.12, let G (s, σ) denote s ∧ σ, and let G_{0} (s, σ) denote s ∧ σ − sσ. Then letting b_{ts}^{A} denote ⟨b_{ts}, A⟩_{𝔎} and X_{ts}^{A} denote ⟨X_{ts}, A⟩_{𝔎} we have:-

1.

b_{dts}^{A}b_{dtσ}^{B} = ⟨A, B⟩_{𝔎} G (s, σ) dt
b_{tds}^{A}b_{τds}^{B} = ⟨A, B⟩_{𝔎} G (t, τ) ds

2.

X_{dts}^{A}X_{dtσ}^{B} = ⟨A, B⟩_{𝔎} G_{0} (s, σ) dt.

3.

EX_{tσ}^{A}b_{ts}^{B} = ⟨A, B⟩_{𝔎} (1 − σ) log (\frac{1}{1 − s ∧ σ} ) dt.

Proof.

Let 𝔊_{t} ≡ σ {b_{rs}|s ∈ [0, 1] , r ∈ [0, t]} .

Then Remark 6.2 implies that

𝔊_{t} = σ {X_{rs}|s ∈ [0, 1] , r ∈ [0, t]} .

Thus by computing correlations, one sees that the increments X_{ts} − X_{τs}, b_{ts} − b_{τs} are independent of 𝔊_{τ} if t > τ. We want to show if two mean-zero 𝔊_{τ} -martingales M_{τ} , N_{τ} have independent increments then

M_{t}N_{t} − EM_{t}N_{t} is a martingale.

Then we shall be able to conclude that

b_{dts}^{A}b_{dtσ}^{B} = d_{t}Eb_{ts}^{A}b_{tσ}^{B} = ⟨A, B⟩_{𝔎} G (s, σ) dt
b_{tds}^{A}b_{τds}^{B} = d_{s}Eb_{ts}^{A}b_{τs}^{B} = ⟨A, B⟩_{𝔎} G (t, τ) ds
X_{dts}^{A}X_{dtσ}^{B} = d_{t}EX_{ts}^{A}X_{tσ}^{B} = ⟨A, B⟩_{𝔎} G_{0} (s, σ) dt
X_{dtσ}^{A}b_{dts}^{B} = d_{t}EX_{ts}^{A}b_{tσ}^{B}.

Let t > s. Then

E (M_{t}N_{t} − EM_{t}N_{t}|𝔊_{s}) = E (M_{t}N_{t}|𝔊_{s}) − EM_{t}N_{t}
= E (M_{t} − M_{s}) (N_{t} − N_{s}) + N_{s}E (M_{t} − M_{s})
+M_{s}N_{s} + M_{s}E (N_{t} − N_{s}) − EM_{t}N_{t}
= M_{s}N_{s} + E (M_{t} − M_{s}) (N_{t} − N_{s}) − EM_{t}N_{t}
= M_{s}N_{s} − EM_{s}N_{s}.

Thus the joint quadratic variation

M_{dt}N_{dt} = d_{t}EM_{t}N_{t}.

It remains only to find X_{dtσ}^{A}b_{dts}^{B} by computing EX_{ts}^{A}b_{tσ}^{B}. But this is just

(8.3)  EX_{tσ}^{A}b_{ts}^{B} = EX_{tσ}^{A}X_{ts}^{B} + ∫_{0}^{s} \frac{EX_{tσ}^{A}X_{tu}^{B}}{1 − u} du
= ⟨A, B⟩_{𝔎} t [G_{0} (s, σ) + ∫_{0}^{s} \frac{G_{0} (u, σ)}{1 − u}du] .

∫_{0}^{s} \frac{G_{0} (u, σ)}{1 − u}du = ∫_{0}^{s} \frac{u ∧ σ}{1 − u}du − ∫_{0}^{s} \frac{uσ}{1 − u}du
= ∫_{0}^{s∧σ} \frac{u}{1 − u}du + ∫_{s∧σ}^{s} \frac{σ}{1 − u}du − ∫_{0}^{s} \frac{uσ}{1 − u}du
= (1 − σ) ∫_{0}^{s∧σ} \frac{u}{1 − u}du + ∫_{s∧σ}^{s} \frac{σ − uσ}{1 − u} du
= (1 − σ) ∫_{0}^{s∧σ} (\frac{1}{1 − u} − 1) du + σ (s − s ∧ σ)
= (1 − σ) log \frac{1}{1 − s ∧ σ} − (1 − σ) (s ∧ σ) + σ (s − s ∧ σ)
= (1 − σ) log \frac{1}{1 − s ∧ σ} − G_{0} (s, σ) .

Returning to Eq.[8.3] yields

EX_{tσ}^{A}b_{ts}^{B} = ⟨A, B⟩_{𝔎} t (1 − σ) log \frac{1}{1 − s ∧ σ} .

Lemma 8.4. b_{ts} is a 𝔎-valued Brownian Sheet starting at 0 such that

Eb_{ts}^{i}b_{τσ}^{j} = δ_{ij}(t ∧ τ)(s ∧ σ) where b_{ts}^{i} ≡ ⟨b_{ts}, A_{i}⟩_{𝔎} where A_{i} runs through an orthonormal basis of 𝔎.

Proof.

Eb_{ts}^{i}b_{τσ}^{j} = E[X_{ts}^{i} + ∫_{0}^{s} \frac{X_{tu}^{i}}{(1 − u)}du][X_{τσ}^{j} + ∫_{0}^{σ} \frac{X_{τu}^{j}}{(1 − u)}du]
= EX_{ts}^{i}X_{τσ}^{j} + EX_{ts}^{i}[∫_{0}^{σ} \frac{X_{τu}^{j}}{(1 − u)}du] + EX_{τσ}^{j}[∫_{0}^{s} \frac{X_{tu}^{i}}{(1 − u)}du]

+E[∫_{0}^{s} \frac{X_{tu}^{i}}{(1 − u)}du][∫_{0}^{σ} \frac{X_{τu}^{j}}{(1 − u)}du]

= I + J + K + L.
I = δ_{ij}(t ∧ τ)(s ∧ σ − sσ).

Now by Tonelli we have ,

E ∫_{0}^{σ} |X_{ts}^{i} \frac{X_{τu}^{j}}{(1 − u)}| du = ∫_{0}^{σ} E |X_{ts}^{i}X_{τu}^{j}| \frac{du}{(1 − u)}
< ∫_{0}^{σ}[E |X_{ts}^{i}|^{2} E |X_{τu}^{j}|^{2}]^{\frac{1}{2}} \frac{du}{(1 − u)}
= ∫_{0}^{σ}[tτ(s − s^{2})(u − u^{2})]^{\frac{1}{2}} \frac{du}{(1 − u)} < ∞.

Therefore by Fubini,

J = E ∫_{0}^{σ} X_{ts}^{i} \frac{X_{τu}^{j}}{(1 − u)}du
= ∫_{0}^{σ} δ_{ij}(t ∧ τ)(s ∧ u − su)\frac{du}{(1 − u)}
= δ_{ij}(t ∧ τ)[∫_{0}^{s∧σ}(1 − s)\frac{udu}{(1 − u)} + ∫_{s∧σ}^{σ} sdu]
= δ_{ij}(t ∧ τ)[(1 − s) ∫_{0}^{s∧σ}(\frac{1}{(1 − u)} − 1)du + s(σ − s ∧ σ)]
= δ_{ij}(t ∧ τ)[(1 − s) log(\frac{1}{1 − s ∧ σ}) − (1 − s)s ∧ σ + s(σ − s ∧ σ)]
= −δ_{ij}(t ∧ τ)[(1 − s) log(1 − s ∧ σ) + s ∧ σ − sσ].

Similarly,

K = −δ_{ij}(t ∧ τ)[(1 − σ) log(1 − s ∧ σ) + s ∧ σ − sσ]

By Tonelli,

E  [∫_{0}^{s} ∫_{0}^{σ} |\frac{X_{tu}^{i}}{(1 − u)} \frac{X_{τν}^{j}}{(1 − ν)}| dudν]
= ∫_{0}^{s} ∫_{0}^{σ} E |\frac{X_{tu}^{i}}{(1 − u)} \frac{X_{τν}^{j}}{(1 − ν)}| dudν
< ∫_{0}^{s} ∫_{0}^{σ} \frac{dudν}{(1 − u)(1 − ν)} (E (X_{tu}^{i})^{2} E (X_{τν}^{j})^{2})^{\frac{1}{2}}
= tτ ∫_{0}^{s} ∫_{0}^{σ}(uν)dudν < ∞.

Therefore by Fubini,

L = E[∫_{0}^{s} ∫_{0}^{σ} \frac{X_{tu}^{i}}{(1 − u)} \frac{X_{τν}^{j}}{(1 − ν)}dudν]
= ∫_{0}^{s} ∫_{0}^{σ} E \frac{X_{tu}^{i}}{(1 − u)} \frac{X_{τν}^{j}}{(1 − ν)}dudν
= ∫_{0}^{s} ∫_{0}^{σ} δ_{ij}(t ∧ τ)(u ∧ ν − uν)\frac{dudν}{(1 − u)(1 − ν)}

= δ_{ij}(t ∧ τ) (∫_{\{s>u>ν>0\}∪\{σ>ν\}} dudν \frac{ν}{1 − ν} + ∫_{\{σ>ν>u>0\}∪\{s>u\}} dudν \frac{u}{1 − u})

= δ_{ij}(t ∧ τ) ∫_{0}^{s∧σ} dν (\frac{ν(s − ν)}{1 − ν} + \frac{ν(σ − ν)}{1 − ν})
= δ_{ij}(t ∧ τ) ∫_{0}^{s∧σ} dν(\frac{1}{1 − ν} − 1)(s + σ − 2ν)
= δ_{ij}(t ∧ τ) [∫_{0}^{s∧σ} \frac{s + σ − 2ν}{1 − ν} dν − ((s + σ)(s ∧ σ) − (s ∧ σ)^{2})]
= δ_{ij}(t ∧ τ) [∫_{0}^{s∧σ} (\frac{s + σ − 2}{1 − ν} + 2) dν − ((s + σ)(s ∧ σ) − (s ∧ σ)^{2})]
= δ_{ij}(t ∧ τ) [∫_{0}^{s∧σ} \frac{s + σ − 2}{1 − ν}dν + ((s ∧ σ)^{2} + 2(s ∧ σ) − (s + σ)(s ∧ σ))]
= δ_{ij}(t ∧ τ) [(s ∧ σ)^{2} + 2(s ∧ σ) − (s + σ)(s ∧ σ) − (s + σ − 2) log(1 − s ∧ σ)] .

So putting it all together, we have

I+J + K + L
= δ_{ij}(t ∧ τ) [(s ∧ σ)^{2} + sσ + (s ∧ σ) − (s + σ)(s ∧ σ)]
= δ_{ij}(t ∧ τ) (σ ∧ s) .

Therefore,

Eb_{ts}^{i}b_{τσ}^{j} = δ_{ij}(t ∧ τ)(s ∧ σ).

b a linear transformation of a Gaussian process and is hence Gaussian. Hence the above assertion is enough to show b is a 𝔎-valued Brownian Sheet.

Theorem 8.5. There exists a Brownian sheet b_{ts} such that X_{ts} can be expressed as:

X_{ts} = b_{ts} − ∫_{0}^{s} b_{tσ} \frac{(1 − s)}{(1 − σ)^{2}} dσ, (8.4)

where b is a 𝔎-valued 2-parameter Brownian Sheet.

Proof. Define b_{ts} ≡ S(X_{t.})(s). Then by Lemma 8.4, b_{ts} is a 𝔎-valued Brownian Sheet. Now by Lemma 8.6, Eq. (8.4) holds.

Lemma 8.6. T : b_{t.} → X_{t.} Proof.

T (b_{t.})(s) = b_{ts} − ∫_{0}^{s} b_{tσ} \frac{(1 − s)}{(1 − σ)^{2}} dσ
= [X_{ts} + ∫_{0}^{t} \frac{X_{tσ}}{(1 − σ)}dσ] − ∫_{0}^{s}[X_{tσ} + ∫_{0}^{σ} \frac{X_{tu}}{(1 − u)}du]\frac{(1 − s)}{(1 − σ)^{2}} dσ
= X_{ts} + ∫_{0}^{s} X_{ts} \frac{(s − σ)}{(1 − σ)^{2}} dσ − (1 − s) ∫_{0}^{s} ∫_{0}^{σ} \frac{X_{tu}}{(1 − u)}dud(\frac{1}{1 − σ})
= X_{ts} + ∫_{0}^{s} X_{tσ} \frac{(s − σ)}{(1 − σ)^{2}} dσ − ∫_{0}^{s} \frac{X_{tu}}{(1 − u)}du + (1 − s) ∫_{0}^{s}(\frac{1}{1 − σ})\frac{X_{tσ}}{(1 − σ)}dσ
= X_{ts} + ∫_{0}^{s} X_{tσ}[\frac{(s − σ)}{(1 − σ)^{2}} − \frac{1}{(1 − σ)} + \frac{1 − s}{(1 − σ)^{2}} ]dσ
= X_{ts}.

Lemma 8.7 (Evaluation of L^{2}-norms). Let M_{m} (ℝ) be as in Remark 2.18. Let {f_{i,A} (·)} be a collection of continuous adapted M_{m} (ℝ)-valued processes. (i.e. f_{i,A} (σ) ∈ 𝔉_{t_{i−1}σ}). Let Δ_{i}b^{A}(σ) ≡ b^{A}(t_{i}, σ)−b^{A}(t_{i−1}, σ) and recall that b^{A} = ⟨b, A⟩_{𝔎} where b is the Brownian sheet from Theorem 3.19. Then

E ∥∑_{i,A} ∫_{0}^{s} f_{i,A}(σ)dΔ_{i}b^{A}(σ)∥_{HS}^{2} = E ∑_{i,A} Δ_{i}t ∫_{0}^{s} ∥f_{i,A}(σ)∥_{HS}^{2} dσ.

Proof. Let (A)_{pq} denote the p, q entry of the matrix A ∈ M_{m} (ℝ). Let

J ≡ E ∥∑_{i,A} ∫_{0}^{s} f_{i,A}(σ)dΔ_{i}b^{A}(σ)∥_{HS}^{2} .

Then

J = ∑_{p,q} E (∑_{i,A} ∫_{0}^{s} (f_{i,A}(σ))_{pq} dΔ_{i}b^{A}(σ))^{2}

= ∑_{p,q} E ∑_{i,A,i′,A′} ∫_{0}^{s} (f_{i,A}(σ))_{pq} (f_{i′,A′}(σ))_{pq} dΔ_{i}b^{A}(σ)dΔ_{i′}b^{A′}(σ)

= ∑_{p,q} E ∑_{i,A} ∫_{0}^{s} Δ_{i}t (f_{i,A}(σ)_{pq})^{2} dσ

= E ∑_{i,A} Δ_{i}t ∫_{0}^{s} ∥f_{i,A}(σ)∥_{HS}^{2} dσ.

8.3. Proof of Lemma 8.8.

Lemma 8.8. Recall that K ⊂ GL_{m} (ℝ) as in Remark 2.18. Let F and G be the exponential and inverse map respectively on matrices as in Definition 6.9. Then the following relations hold where A ∈ U and B, C ∈ 𝔎:-

ON THE EQUIVALENCE OF MEASURES ON LOOP SPACE 89

1. F′(A)B = ∫_{0}^{1} F [(1 − τ)A] BF [τA] dτ. (8.5)

2.

F′′(A)B ⊗ C = ∫_{0}^{1} dτ ∫_{0}^{1}(1 − u)F [(1 − τ) (1 − u) A]
×CF [τ(1 − u)A] BF [uA] du

+ ∫_{0}^{1} dτ ∫_{0}^{1} uF [(1 − u)A] B (8.6)

×F [(1 − τ)uA] CF [τuA] du.

3. G′(A)B = −A^{−1}BA^{−1}. (8.7)

4. G′′(A)B ⊗ C = A^{−1}BA^{−1}CA^{−1} + A^{−1}CA^{−1}BA^{−1}. (8.8)

5.

sup_{A∈𝔎} ∥F′(A)B∥_{HS} ≤ Const ∥B∥_{HS} . (8.9)

6.

sup_{A∈𝔎} ∥F^{(n)}(A)B_{1} ⊗ ⋯ ⊗ B_{n}∥_{HS} ≤ Const ∥B_{1}∥_{HS} ⋯ ∥B_{n}∥_{HS} . (8.10)

Proof.

\frac{d}{dt}F (t (A + sB)) = \frac{d}{dt}e^{t(A+sB)} = (A + sB) e^{t(A+sB)}.

\frac{d}{ds} \frac{d}{dt}F (t (A + sB)) |_{s=0} = Be^{tA} + tA (F′ (tA) tB) .

\frac{d}{dt} [\frac{d}{ds}F (t (A + sB)) |_{s=0}] = Be^{tA} + A [\frac{d}{ds}F (t (A + sB)) |_{s=0}] .

By Duhammel’s Principle (or method of integrating factors) we get Eq. (8.5)

\frac{d}{dt} \{e^{−tA} [\frac{d}{ds}F (t (A + sB)) |_{s=0}]\} = e^{−tA}Be^{tA}.

\frac{d}{ds}F (A + sB) |_{s=0} = e^{A} ∫_{0}^{1} e^{−tA}Be^{tA}dt.

\frac{d}{ds}F (A + sB) |_{s=0} = e^{A} ∫_{0}^{1} e^{−tA}Be^{tA}dt.

F′′ (A) B ⊗ C = \frac{d}{dt}F′(A + tC)B|_{t=0}
= ∫_{0}^{1} \frac{d}{dt}F [(1 − u) (A + tC)] BF [u (A + tC)] |_{t=0}du
= ∫_{0}^{1}(1 − u) (F′ [(1 − u) A] C) Be^{uA}du
+ ∫_{0}^{1} ue^{(1−u)A}BF′ (uA) Cdu,

which is Eq. (8.6).

(A + tB) G (A + tB) = 1
BG (A) + A (G′(A)B) = 0
G′(A)B = −G(A)BG(A),

which is Eq. (8.7).

G′′ (A)B ⊗ C

= \frac{d}{dt}G′(A + tC)B|_{t=0}

= −\frac{d}{dt}G(A + tC)BG(A + tC)|_{t=0}

= G(A)CG(A)BG(A) + G(A)BG(A)CG(A),

which is Eq. (8.8).

It will not be necessary to prove Eq. (8.9) if we can show Eq. (8.10). Let B̃_{i} ≡ B_{i}/ ∥B_{i}∥_{HS}

F^{(n)}(A)B_{1} ⊗ ⋯ ⊗ B_{n}
= ∥B_{1}∥_{HS} ⋯ ∥B_{n}∥_{HS} F^{(n)}(A)B̃_{1} ⊗ ⋯ ⊗ B̃_{n}.

Now

sup_{A,B̃_{1}···B̃_{n}} ∥F^{(n)}(A)B̃_{1} ⊗ ⋯ ⊗ B̃_{n}∥_{HS}

is the supremum of a continuous function over a compact set and is hence finite. Call this supremum C. Therefore,

sup_{A∈𝔎} ∥F^{(n)}(A)B_{1} ⊗ ⋯ ⊗ B_{n}∥_{HS} ≤ C ∥B_{1}∥_{HS} ⋯ ∥B_{n}∥_{HS} ,

which is Eq. (8.10).

8.4. Gaussian Measures. For further information on this topic see Kuo [[23]].

Definition 8.9 (Gaussian Measure). Ω is a separable Banach Space with |·|_{Ω}. μ a (mean-zero, non-degenerate) Gaussian measure on Ω iff

μ̂ (ϕ) ≡ ∫ μ (dx) exp iϕ (x) = exp [−\frac{1}{2}q (ϕ, ϕ)] ,

and q : Ω^{∗} × Ω^{∗} → ℝ an inner product on Ω^{∗}. Ω^{∗} denotes the dual to Ω; i.e. the set of all bounded linear maps from Ω to ℝ. μ̂ denotes the Fourier transform of the measure μ.

Remark 8.10. An alternate (and equivalent definition) of a Gaussian measure μ on a Banach space Ω is a measure so that any ψ ∈ Ω^{∗} (an ℝ-valued random variable) is a mean-zero normal random variable. One can see that if μ satisfies Definition 8.9 then it satisfies the alternate Definition in Remark 8.10. This is because for any ψ ∈ Ω^{∗},

{ \atop Law_{ψ}} (λ) ≡ Law_{ψ} exp iλx = μ exp iλψ (x) = exp −\frac{λ^{2}}{2} q (ψ, ψ) ,

which means that ψ has a normal distribution with mean 0 and variance q (ψ, ψ). (Although this second definition appears weaker, it is possible to prove Definition 8.9 from it.)

We shall repeatedly use the following well-known Theorem due to Fernique:

Theorem 8.11 (Fernique). Let μ be a Gaussian measure on a Banach Space Ω. Then there exists an ε > 0 so that

μ [exp ε |x|_{Ω}^{2}] < ∞.

(Note:- this also means that μ |x|_{Ω}^{2} < ∞)

Lemma 8.12 (Bochner Integrals). Define

S ≡ {f : Ω → Ω| Ran f is a finite set} .

Let L^{1} (μ, Ω) denote {f : Ω → Ω| ∫ |f (x)|_{Ω} μ (dx) < ∞}. Then there exists a linear functional I : L^{1} (μ, Ω) → Ω with the following properties:-

1.

I (f) = ∑_{x∈Ω} xμ (f^{−1} {x}) if f ∈ S.

2.

|I (f)|_{Ω} ≤ ∫ |f|_{Ω} μ (dx) = ∥f∥_{L^{1}(μ,Ω)} .

3.

ψ (I (f)) = ∫ ψ (f) μ (dx) for any ψ ∈ Ω^{∗}.

Henceforth we shall write ∫ f (x) μ (dx) in place of the less intuitive I (f).

Proof. First we show that S is dense in L^{1} (μ, Ω). We shall use the separability of Ω to do this. Let {x_{n}} be a countable dense subset of Ω. Cover Ω with measurable sets B_{i} as follows B_{1} ≡ B (ε, x_{1}) , ⋯, B_{n} ≡ B (ε, x_{n}) − ∪_{i=1}^{n}B (ε, x_{i}) , · · ·. Given f ∈ L^{1} (μ, Ω), let ϕ_{ε} ≡ ∑_{i} x_{i}1_{f^{−1}(B_{i})}. Then

∫ |f − ϕ_{ε}|_{Ω} μ (dx) ≤ ∑_{i} ∫ |f − x_{i}|_{Ω} 1_{f^{−1}(B_{i})}μ (dy) ≤ εμ (Ω) .

Since the ϕ_{ε} were all in S, S is dense in L^{1} (μ, Ω).

Define I (f) = ∑_{x∈Ω} xμ (f^{−1} {x}) on S. Hence property 1. is already satisfied.

Notice that I is a linear functional on S. Secondly, if ψ ∈ Ω^{∗}, on S

ψ (I (f)) = ∑_{x∈Ω} ψ (x) μ (f^{−1} {x})

= ∫ μ (dy) ∑_{x∈Ω} ψ (x) 1_{f^{−1}{x}} (y)

= ∫ ψ (∑_{x∈Ω} x1_{f^{−1}{x}}) dμ

= ∫ ψ ∘ f (y) μ (dy) .

Also

|I (f)|_{Ω} ≤ ∑_{x∈Ω} |x|_{Ω} μ (f^{−1} {x}) = ∥f∥_{L^{1}(μ,Ω)} .

Extend our definition of I to L^{1} (μ, Ω) by defining I (f) ≡ lim_{n→∞} I (f_{n}) whenever f_{n} → f in L^{1} (μ, Ω) with {f_{n}} ∈ S.

Property 1 holds by definition. The linearity of I holds despite the extension. Property 2 holds as well since

|I (f)|_{Ω} = |lim I (f_{n})|_{Ω} = lim |I (f_{n})|_{Ω} < lim ∥f_{n}∥_{L^{1}(μ,Ω)} = ∥f∥_{L^{1}(μ,Ω)} .

If ψ ∈ Ω^{∗} we have f_{n} → f in L^{1} (μ, Ω). This means I (f_{n}) → I (f) in Ω. Thus ∫ ψ ∘ f_{n}dμ = ψ ∘ I (f_{n}) → I (f) in Ω.However,

|∫ ψ (f) dμ − ∫ ψ (f_{n}) dμ|_{Ω} ≤ ∫ |ψ (f) − ψ (f_{n})|_{Ω} dμ
≤ |ψ|_{Ω^{∗}} ∥f − f_{n}∥_{L^{1}(μ,Ω)}
→ 0.

Thence we have ∫ ψ ∘ fdμ = I (f) and Property 3 holds.

Theorem 8.13 (Cameron-Martin space). We construct H, the Cameron-Martin space associated with the Gaussian measure μ:-

1. Ω^{∗} ⊂ L^{p} (μ) for all p.

2. There is a linear map J : Ω^{∗} → Ω so that ϕ (Jψ) = ⟨ψ, ϕ⟩_{L^{2}(μ)} .

3. q (ϕ, ψ) = ∫ ϕ (x) ψ (x) μ (dx) .

4. H ≡ {x ∈ Ω : sup_{ϕ∈Ω^{∗}} |ϕ (x)|^{2} /q (ϕ, ϕ) < ∞}is a subspace of Ω.

5. |h|_{Ω} ≤ const |h|_{H} for any h ∈ H.

6. Let K be the closure of Ω^{∗} in L^{2} (μ). Then there exists an extension of the map J from K → H so that J : f → ∫ xf (x) μ (dx). Furthermore, J is an isometry onto H and H is dense in Ω. In particular, H is a Hilbert space under the isometry from K.

7. Letting h run through an orthonormal basis of H,

∫ ϕ (x) ψ (x) μ (dx) = ∑ ψ (h) ϕ (h) .

Proof. Proceeding in order, we prove:- 1. If ϕ ∈ Ω^{∗} then

∫ |ϕ (x)|_{Ω}^{p} μ (dx) ≤ |ϕ|_{Ω^{∗}}^{p} ∫ |x|_{Ω}^{p} μ (dx) < ∞

by Theorem 8.11 (Fernique).

2. Define J (ϕ) ≡ ∫ xϕ (x) μ (dx) for any ϕ ∈ Ω^{∗}. If ψ ∈ Ω^{∗}, then by property 3 of Lemma 8.12

ψ (J (ϕ)) = ∫ ψ (x) ϕ (x) μ (dx) = ⟨ψ, ϕ⟩_{L^{2}(μ)} .

3. By Definition 8.9, we have

μ̂ (tϕ) = μ (exp itϕ (x)) = exp −t^{2} [\frac{1}{2}q (ϕ, ϕ)] .

If we can show [∂_{t}∂_{t}μ̂ (tϕ)] ↓_{t=0}= −μϕ (x)^{2} we would have

μϕ (x)^{2} = q (ϕ, ϕ)

by taking two derivatives on the right hand side. Then

μ [ϕψ] = \frac{1}{4}μ (ϕ + ψ)^{2} − \frac{1}{4}μ (ϕ − ψ)^{2} = q (ϕ, ψ) ,

and we would be done. So the problem reduces to computing ∂_{t}∂_{t}μ̂ (tϕ). We shall show

∂_{t}∂_{t}μ̂ (tϕ) = μ [−ϕ (x)^{2} exp itϕ (x)]

by Dominated Convergence and Theorem 8.11 as follows:

∂_{t}μ̂ (tϕ) = lim_{ε↓0} μ (\frac{exp i (t + ε) ϕ (x) − exp itϕ (x)}{ε})
= lim_{ε↓0} μ (\frac{exp iεϕ (x) − 1}{ε}) exp itϕ (x) .

Now |exp itϕ (x)| < 1 so to apply dominated convergence, it will suffice to dominate \frac{exp iεϕ(x)−1}{ε} by an L^{1} (dμ) function.

|\frac{exp iεϕ (x) − 1}{ε}| = \frac{1}{ε} ∑_{n>1} |iεϕ (x)|^{n} /n! ≤ |ϕ (x)| ∑_{n>0} |εϕ (x)|^{n} / (n + 1)!
≤ |ϕ (x)| exp ε |ϕ|_{Ω^{∗}} |x|_{Ω} .

By Theorem 8.11 (Fernique), we have

μ |ϕ (x)| exp ε |ϕ|_{Ω^{∗}} |x|_{Ω} ≤ \sqrt{μ |ϕ|^{2} ∥x∥_{Ω}^{2}}\sqrt{μ exp (ε |ϕ|_{Ω^{∗}})^{2} |x|_{Ω}^{2}} < ∞.

Thus by dominated convergence, the limit goes through and we have

∂_{t}μ̂ (tϕ) = μ (iϕ (x) exp itϕ (x)) .

Similarly to take the second derivative, we have to again verify that the limit can be passed through the integral in Eq.(8.11) below.

(8.11) ∂_{t}∂_{t}μ̂ (tϕ) = lim_{ε↓0} μ (iϕ (x) exp itϕ (x) [\frac{exp iεϕ (x) − 1}{ε}]) .

Now using Fernique yet again, iϕ (x) [\frac{exp iεϕ(x)−1}{ε}] is bounded by the L^{1} (dμ)

function |ϕ (x)|^{2} exp ε |ϕ|_{Ω^{∗}} |x|_{Ω}. Thus by dominated convergence, we have

∂_{t}∂_{t}μ̂ (tϕ) = μ [−ϕ (x)^{2} exp itϕ (x)] .

Hence we have shown part 3.

4. If h, k ∈ H, then

\begin{matrix}| & ϕ & (αh + k) & | \\ √ & \overline{q (ϕ, ϕ)} \end{matrix} ≤ |α| \begin{matrix}| & ϕ & (h) & | \\ √ & \overline{q (ϕ, ϕ)} \end{matrix} + \begin{matrix}| & ϕ & (k) & | \\ √ & \overline{q (ϕ, ϕ)} \end{matrix},

which implies that H is a subspace of Ω.

5. By Fernique’s Theorem (Theorem 8.11)∫ μ (dx) |x|_{Ω}^{2} ≤ ∫ μ (dx) exp ε |x|_{Ω}^{2} < ∞. As a consequence,

q (ϕ, ϕ) = ∫ μ (dx) ϕ (x)^{2}
≤ |ϕ|_{Ω^{∗}}^{2} ∫ μ (dx) |x|_{Ω}^{2}
= C^{2} |ϕ|_{Ω^{∗}}^{2} .

Therefore

|h|_{Ω} = sup_{ϕ∈Ω^{∗}} \begin{matrix}| & ϕ & (x) & | \\ |ϕ| & Ω_{∗} \end{matrix}
≤ C sup_{ϕ∈Ω^{∗}} \frac{|ϕ (x)|}{q (ϕ, ϕ)}
≤ C |h|_{H} .

6. Clearly, Ω^{∗} ⊂ H. Let {ψ_{n}} be Cauchy in L^{2} (dμ). Then

|Jψ_{n} − Jψ_{m}| ≤ μ |x|_{Ω} |(ψ_{n} − ψ_{m}) (x)|
≤ (μ |x|_{Ω}^{2}) μ (ψ_{n} − ψ_{m})^{2} → 0 as n,m go to ∞.

Thus by completeness of Ω, {Jψ_{n}} converges in Ω. Thus we extend the map

J to the space K ≡ \overline{Ω^{∗}}^{L^{2}(dμ)} by continuity.

Let x ∈ Im J. Then there is some sequence {Jψ_{n}} so that ψ_{n} ∈ Ω^{∗} , ψ_{n} converges in L^{2} (dμ) to some ψ ∈ K. But then

sup_{ϕ∈Ω^{∗}} |ϕ (x)|^{2} /q (ϕ, ϕ) = sup_{ϕ∈Ω^{∗}} lim_{n→∞} (μ [ϕψ_{n}] /\sqrt{μ [ϕ^{2}]})^{2}

= sup_{ϕ∈Ω^{∗}} (μ [ϕψ] /\sqrt{μ [ϕ^{2}]})^{2}
= μ [ψ^{2}] .

Thus Im J ⊂ H, and J : K → Im H is an isometry.

Im J dense in Ω. If not there’s a non-trivial ψ ∈ Ω^{∗} so that ψ (Im J) = 0. This means that μ (ψ^{2}) = 0 which is a contradiction. Thus Im J and hence also H are dense in Ω.

Let h ∈ H. Let ĥ (ϕ) ≡ ϕ (h) for any ϕ ∈ Ω^{∗}. Note that |ĥ (ϕ)| = |ϕ (h)| < |ϕ|_{L^{2}(dμ)} |h|_{H}. Thus h ∈ K^{∗}. Thus there exists an f ∈ K so that ϕ (h) = μ [ϕf] for any ϕ ∈ Ω^{∗}. But now ϕ (Jf) = μ [ϕ (x) f (x)] = ϕ (h). Thus h = Jf ⇒ H ⊂ Im J. Thus J is a unitary map from K to H and in particular, H is a Hilbert space that’s dense in Ω.

7. Let h run through an orthonormal basis of H. Then

∫ ϕ (x) ψ (x) μ (dx) = ⟨J_{ϕ}, J_{ψ}⟩_{H}
= ∑ ⟨J_{ϕ}, h⟩_{H} ⟨h, J_{ψ}⟩_{H}
= ∑ ϕ (h) ψ (h) .

Example 8.14 (Computing Cameron-Martin Spaces). Let Ω be the Banach space

L (𝔎) ≡ {x ∈ C ([0, 1] → 𝔎) |x (0) = x (1) = 0}

equipped with the uniform norm. μ = LawX_{t·}.(X_{t·} is a Brownian bridge from 0 to 0 with parameter t). Let H_{0,t} denote the Cameron-Martin space associated to the measure μ. Then H_{0,t} is the space H_{0} (𝔎) equipped with the inner product

⟨k (·) , l (·)⟩_{H_{0,t}} = \frac{1}{t} ∫_{0}^{1} ⟨k′ (u) , l′ (u)⟩_{𝔎} du.

Recall from Definition 2.1 that

H_{0} (𝔎) = {x ∈ Ω : x has one L^{2} ([0, 1] , dλ) -derivative} .

Proof. X_{t·} is a Brownian bridge with parameter t. So μ = LawX_{t·} is already a Gaussian measure. Furthermore,

∫ μ (dx) ⟨x (s) , A⟩_{𝔎} ⟨x (σ) , B⟩_{𝔎}
= E ⟨X_{ts}, A⟩_{𝔎} ⟨X_{τσ}, B⟩_{𝔎}
= ⟨A, B⟩_{𝔎} tG_{0} (s, σ) .

Define an element ψ_{A,s} of Ω^{∗} by setting ψ_{A,s} (x) = ⟨x (s) , A⟩_{𝔎}. Then

q (ψ_{A,s}, ψ_{B,σ}) = E ⟨X_{ts}, A⟩_{𝔎} ⟨X_{τσ}, B⟩_{𝔎} = ⟨A, B⟩_{𝔎} tG_{0} (s, σ) .

Let J be the standard inclusion of Ω^{∗} into H_{0,t} as in Theorem 8.13. Then from abstract nonsense (i.e. Theorem 8.13)

⟨(Jψ_{A,s}) (σ) , B⟩_{𝔎} = ψ_{B,σ} (Jψ_{A,s})
= q (ψ_{B,σ}, ψ_{A,s})
= ⟨AtG_{0} (s, σ) , B⟩_{𝔎} .

Therefore Jψ_{A,s} = AtG_{0} (s, ·). Let ω ∈ H_{0,t}. ω ⊥ Jψ_{A,s} ⟺ ⟨ω (s) , A⟩_{𝔎} = 0. Therefore,

Λ ≡ span ⟨Jψ_{A,s} : s ∈ (0, 1) , A ∈ 𝔎⟩ is dense in H_{0,t}.

Thus it will suffice for us to specify the norm of H_{0,t} on Λ.

\frac{1}{t} ∫_{0}^{1} ⟨(Jψ_{B,σ})′ (u) , (Jψ_{A,s})′ (u)⟩_{𝔎} du

= \frac{1}{t} ∫_{0}^{1} ⟨\frac{d}{du}tG_{0} (u, σ) B, \frac{d}{du}tG_{0} (u, s) A⟩_{𝔎} du

= t ⟨B, A⟩_{𝔎} ∫_{0}^{1} (1_{{u≤σ}} − σ) (1_{{u≤s}} − s) du

= t ⟨B, A⟩_{𝔎} [s ∧ σ − sσ − σs + sσ]

= t ⟨B, A⟩_{𝔎} G_{0} (s, σ)

= q (ψ_{A,s}, ψ_{B,σ}) .

Thus the inner product \frac{1}{t} ∫_{0}^{1} ⟨k′ (u) , l′ (u)⟩_{𝔎} du works on Λ and H_{0,t} is the closure of Λ under this norm.

Define

H̃ ≡ {x ∈ Ω : x has one L^{2} ([0, 1] , dλ) -derivative} .

We want to show H̃ = H_{0,t}.

Let y ∈ C^{∞} [0, 1] so that y (0) = y (1) = 0. Define ψ_{y} ∈ Ω^{∗} as follows:

ψ_{y} : x ↦ −\frac{1}{t} ∫_{0}^{1} ⟨y′′ (u) , x (u)⟩_{𝔎} du.

Also

⟨Jψ_{y} (s) , A⟩_{𝔎} = ψ_{y} (Jψ_{A,s}) = −\frac{1}{t} ∫_{0}^{1} ⟨y′′ (u) , t (s ∧ u − su) A ⟩_{𝔎} du
= − ∫_{0}^{1} ⟨y, A ⟩′′_{𝔎} (u) (s ∧ u − su) du
= − ∫_{0}^{1} (s ∧ u − su) d_{u} ⟨y, A ⟩′_{𝔎}
= ∫_{0}^{1} (1_{{u≤s}} − s) ⟨y, A ⟩′_{𝔎} (u) du
= ⟨y (s) , A ⟩_{𝔎} .

Thus Jψ_{y} = y which implies that smooth loops are in Im J and hence in H_{0,t}. Let x ∈ H̃. Then let y_{n} be smooth so that y_{n} → x′ in L^{2}. Then

ỹ_{n} ≡ y_{n} − ∫_{0}^{1} y_{n} (u) du

also converges to x′ in L^{2}, since

∫_{0}^{1} y_{n} (u) du → ∫_{0}^{1} x′ (u) du = 0.

But then ∫_{0}^{·} ỹ_{n} converge to x in the norm \frac{1}{t} ∫_{0}^{1} ⟨k′ (u) , l′ (u)⟩_{𝔎} du. Thus H̃ ⊂ H_{0,t}. Since H̃ is complete, with respect to the inner product \frac{1}{t} ∫_{0}^{1} ⟨k′ (u) , l′ (u)⟩_{𝔎} du, we have H̃ = H_{0,t} and we are done.

9. Conjectures on Gaussian measure equivalence

Let g be the C ([0, 1] → L (K))-valued random variable so that g (t) is the loop g_{t} of Definition 2.22. Let P_{σ} be the law of σg a measure on C ([0, 1] → L (K)). Then for any probability μ on L (K) we define a measure P_{μ} on C ([0, 1] → L (K)) by

P_{μ} [f] ≡ ∫ μ (dσ) P_{σ} [f] .

Conjecture 1. Let the energy, Ẽ, of an absolutely continuous loop be given by

Ẽ (σ_{[g_{0}]}) ≡ ∫_{0}^{1} |σ_{[g_{0}]} (ds) σ_{[g_{0}]} (s)^{−1}|_{𝔎}^{2} .

Let σ_{[g_{0}]} be an Energy-minimizing loop in the homotopy class [g_{0}]. Recall the μ_{0,t}-a.s. function V_{t} of Theorem 4.1;

V_{t} (x) = \frac{1}{2t^{2}} |∫_{0}^{1} x (ds) x (s)^{−1}|_{𝔎}^{2} − (\frac{dim 𝔎}{2t} + ∂_{t} log P_{t}^{K} (e)) .

Then if f : L (K) → ℝ is “nice” it is reasonable to expect

μ_{0,T} [f] = ∑_{[g_{0}]} ∫ f (x_{T}) exp [lim_{ε→0} ∫_{ε}^{T} V_{t} (x_{t}) dt − \frac{Ẽ (σ_{[g_{0}]})}{2ε}] ν_{σ_{[g_{0}]}} (dx) .

where the [g_{0}] run through all the homotopy classes of K.

Proof. Let μ_{0,t} = ∫ M_{t}Dx where Dx is “Lebesgue measure” on L (K). Let g_{·} be an L (K)-valued Brownian motion. Then

∫ f (∂_{t}M_{t}) Dx

= ∂_{t} ∫ fM_{t}Dx

= ∫ M_{t} (\frac{1}{2}△_{L(K)}f + V_{t}f) Dx

= ∫ f (\frac{1}{2}△_{L(K)} + V_{t}) M_{t}Dx.

Thus M_{t} “satisfies”

∂_{t}M_{t} = (\frac{1}{2}△_{L(K)} + V_{t}) M_{t}.

Working in this vein we have

∂_{t} [M_{T −t} (g_{t}) exp ∫_{0}^{t} V_{T −τ} (g_{τ}) dτ]

= − ((\frac{1}{2}△_{L(K)} + V_{T −t}) M_{T −t}) (g_{t}) exp (∫_{0}^{t} V_{T −τ} (g_{τ}) dτ) dt

+ dtV_{T −t} (g_{t}) M_{T −t} (g_{t}) exp (∫_{0}^{t} V_{T −τ} (g_{τ}) dτ) dt

+ \frac{1}{2}△_{L(K)}M_{T −t} (g_{t}) exp (∫_{0}^{t} V_{T −τ} (g_{τ}) dτ) dt

+ dmartingale
= dmartingale.

Therefore, Let σ_{[g_{0}]} denote the energy-minimizing path in the homotopy class of

[g_{0}]. Let Ẽ (x) denote the energy ∫_{0}^{1} |x (ds) x (s)^{−1}|_{𝔎}^{2} of a path in L (K) (defined

only for absolutely continuous paths with one L^{2}-derivative). Then we have

EM_{T} (g_{0})

= E lim_{ε→0} M_{ε} (g_{T}) exp ∫_{ε}^{T} V_{T −τ} (g_{τ}) dτ

= ∫ exp \frac{−Ẽ (σ_{[g_{0}]})}{2ε}δ_{σ_{[g_{0}]}} (x_{T}) exp [∫_{0}^{T} V_{T −τ} (x_{τ}) dτ] ν_{g_{0}} (dx)

= ∫ δ_{σ_{[g_{0}]}} (x_{T}) exp [lim_{ε→0} ∫_{ε}^{T} V_{T −τ} (x_{τ}) dτ − \frac{Ẽ (σ_{[g_{0}]})}{2ε}] ν_{g_{0}} (dx)

= ∫ δ_{σ_{[g_{0}]}} (x_{T}) δ_{g_{0}} (x_{0}) exp [lim_{ε→0} ∫_{ε}^{T} V_{T −τ} (x_{τ}) dτ − \frac{Ẽ (σ_{[g_{0}]})}{2ε}] ν (dx)

= ∫ δ_{σ_{[g_{0}]}} (x_{0}) δ_{g_{0}} (x_{T}) exp [lim_{ε→0} ∫_{ε}^{T} V_{T −τ} (x_{T −τ}) dτ − \frac{Ẽ (σ_{[g_{0}]})}{2ε}] ν (dx)

= ν_{σ_{[g_{0}]}} [δ_{g_{0}} (x_{T}) exp [lim_{ε→0} ∫_{ε}^{T} V_{t} (x_{t}) dt − \frac{Ẽ (σ_{[g_{0}]})}{2ε}]] .

Here we have used the fact that backwards Brownian motion has the same law as a forwards Brownian motion. σ_{[g_{0}]} has to be homotopic to g_{0} because g_{·} explicitly describes such a homotopy.

ON THE EQUIVALENCE OF MEASURES ON LOOP SPACE 99

Now letting [g_{0}] run through the homotopy classes, we have

μ_{0,T} f

= ∫ f (g_{0}) M_{T} (g_{0}) Dg_{0}

= ∑_{[g_{0}]} ∫ f (g_{0}) δ_{g_{0}} (x_{T}) exp [lim_{ε→0} ∫_{ε}^{T} V_{t} (x_{t}) dt − \frac{Ẽ (σ_{[g_{0}]})}{2ε}] ν_{σ_{[g_{0}]}} (dx) Dg_{0}

= ∑_{[g_{0}]} ∫ f (x_{T}) exp [lim_{ε→0} ∫_{ε}^{T} V_{t} (x_{t}) dt − \frac{Ẽ (σ_{[g_{0}]})}{2ε}] ν_{σ_{[g_{0}]}} (dx) ,

which is the desired result. Example 9.1 (The S^{1} case). Define a measure ν̃_{T} by setting

ν̃_{T} ≡ ∑_{α∈ℤ} C_{α,T} ν_{T}^{S^{1}} (σ_{α}, ·)

where

C_{α,T} ≡ \frac{P_{T}^{ℝ} (0)}{P_{T}^{S^{1}} (e)} exp (−\frac{1}{2T} α^{2}) .

Proof. Let σ_{α} ≡ (cos 2παs, sin 2παs) be the energy-minimizing geodesic in the α^{th} homotopy class for any α ∈ ℤ. Then for any loop x homotopic to σ_{α}, we have

Ẽ (σ_{α}) = ∫_{0}^{1} |α|^{2} ds = α^{2}.

and

V_{t} (x) = \frac{1}{2t^{2}} |∫_{0}^{1} x (ds) x (s)^{−1}|_{𝔎}^{2} − (\frac{1}{2t} dim 𝔎 + ∂_{t} log p_{t}^{K} (e))
= \frac{α^{2}}{2t^{2}} − (\frac{1}{2t} + ∂_{t} log p_{t}^{K} (e)) .

This implies

lim_{ε→0} ∫_{ε}^{T} V_{t} (x_{t}) dτ − \frac{Ẽ (σ_{[g_{0}]})}{2ε}

= lim_{ε→0} ∫_{ε}^{T} \frac{α^{2}}{2t^{2}} − (\frac{1}{2t} + ∂_{t} log p_{t}^{K} (e)) dt − \frac{α^{2}}{2ε}

= lim_{ε→0} \frac{α^{2}}{2ε} − \frac{α^{2}}{2T} − ∫_{ε}^{T} (\frac{1}{2t} + ∂_{t} log P_{t}^{S^{1}} (e)) dt − \frac{α^{2}}{2ε}

= lim_{ε→0} \frac{−α^{2}}{2T} − ∫_{ε}^{T} (\frac{1}{2t} + ∂_{t} log P_{t}^{S^{1}} (e)) dt

= lim_{ε→0} \frac{−α^{2}}{2T} − ∫_{ε}^{T} (∂_{t} log \sqrt{t}P_{t}^{S^{1}} (e)) dt

= \frac{−α^{2}}{2T} − log \sqrt{T} P_{T}^{S^{1}} (e) + log \sqrt{ε}P_{ε}^{S^{1}} (e) .

By Lemma 5.3,

lim_{ε→0} ε^{d/2}P_{ε}^{K} (e) = (2π)^{−d/2} ,

and so

lim_{ε→0} ∫_{ε}^{T} V_{t} (x_{t}) dτ − \frac{Ẽ (σ_{[g_{0}]})}{2ε} = \frac{−α^{2}}{2T} − log \sqrt{T} P_{T}^{S^{1}} (e) − log \sqrt{2π}.

Thus we are done since

μ_{0,T} f = ∑_{[g_{0}]} ∫ f (x_{T}) exp [lim_{ε→0} ∫_{ε}^{T} V_{t} (x_{t}) dt − \frac{Ẽ (σ_{[g_{0}]})}{2T ε}] ν_{σ_{[g_{0}]}} (dx)

= ∑_{α} ∫ f (x_{T}) exp [−\frac{α^{2}}{2T} − log \sqrt{T} P_{T}^{S^{1}} (e) − log \sqrt{2π}] ν_{σ_{α}} (dx)

= ∑_{α} \frac{P_{T}^{ℝ} (0)}{P_{T}^{S^{1}} (e)} ∫ f (x_{T}) exp [−\frac{α^{2}}{2T} ] ν_{σ_{α}} (dx) .

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