ON THE EQUIVALENCE OF MEASURES ON LOOP SPACE

VIKRAM 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 1 2. Statement of Results 3 2.1. Loop group Geometry 3 2.2. Measures on the Loop group 4 2.3. The stochastic framework 7 3. Motivation for Theorem 2.20 8 3.1. Semimartingale Properties of X_{ts} 10 4. Semi-Martingale Properties of g_{T,·} 14 5. HKM ↓_{𝔊_{z}}∽ P W M ↓_{𝔊_{z}} 17 5.1. Proof of Theorem 5.3 19 References 22

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 [25], [17], [23],[3], [13], [14], [1], [20], [16], [11] and the references therein.

Heat Kernel and pinned Wiener measure are two natural measures that have been advocated as the “right” measure on 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 [15], [21], [2], [22]. Heat Kernel measure has been studied in [12], [10] as another natural measure on Loop Space. In [12], 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.

†This research was partially supported by NSF Grant DMS 96-12651.

A further motivation comes from logarithmic Sobolev inequalities and the papers of Getzler [14], Gross [15], Driver [10], 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 [7], [8] for surveys and [18]). 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 [14], 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 [15], 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 [12] 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 [18])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 [12] to eliminate the Gross’ potential term and thereby prove a logarithmic Sobolev inequality for pinned Wiener measure.

In Section 5 we show that pinned Wiener measure is equivalent to Heat Kernel measure on 𝔉_{s}, the σ-algebra 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 [12], 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 4, using extensively the two-parameter calculus developed by Norris in [24], we show that in the other parameter this process is a Brownian semimartingale on the path space of the Lie group. The fact that we can pull back this process to a Lie algebra valued Brownian Semimartingale together with 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.

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. 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 commutative Lie group 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 (The Laplacian △_{L(K)} and △_{L(𝔎)}). Take an 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}.

Remark 2.3 (Motivation for defintion 2.2). In analogy with finite-dimensional Riemannian geometry, given a function f on L (K), we should expect the Laplacian △_{L(K)}f to be div (∇f) where the divergence div ≡ tr∇, with ∇ being the Levi-Civita covariant derivative. So given an orthonormal basis {h} of H_{0} (𝔎), and letting ∂_{h}, in T L (K), be the left-invariant vector field associated to h, we should expect

“△f = div (∇f) = ∑_{h} ∂_{h} · ∇_{∂_{h}} (∇f) = ∑_{h} ∂_{h}^{2}f − (∇_{∂_{h}}∂_{h}) · ∇f.”

However, the sum ∑_{h} ∂_{h}^{2}f − (∇_{∂_{h}}∂_{h}) · ∇f is not well-defined independent of orthonormal basis. As Driver and Lohrenz showed in [12] ∑_{h} ∂_{h}^{2}f − (∇_{∂_{h}}∂_{h}) · ∇f is defined independent of good orthonormal bases. Here an orthonormal basis {h} is good if the Lie bracket [h (s) , h′ (s)] = 0 s-a.s. In that case ∑_{h} ∂_{h}^{2} − (∇_{∂_{h}}∂_{h}) reduces to ∑ ∂_{h}^{2} (which is independent of any orthonormal basis).

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 2.4 (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} (x^{−1}y) 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 2.5 (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 2.6 (Brownian motion on K). We will state two 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 measure 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

d_{s} (ϕ ∘ β) = (ϕ′ ∘ β (s)) d_{s}β + \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 while d_{s}β denotes the Ito differential of β in the s variable.

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 Definition 2.11.

Definition 2.7 (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}^{−1})}{P_{t}^{K} (e)} ∏_{i=1}^{n} P_{t(s_{i}−s_{i−1})}^{K} (x_{i−1}^{−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}.]

Definition 2.8 (Brownian bridge on K). s → χ (s) is a Brownian bridge 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 2.9 (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 meanzero 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 2.10. It turns out that if χ is a Brownian bridge sheet on 𝔎 then χ_{ts} has a version which is continuous in both its parameters, t → χ_{ts} is a Brownian motion on 𝔎 with parameter G_{0} (s, s) and s → χ_{ts} is a Brownian bridge on 𝔎 with parameter t. We will always choose such a jointly-continuous version of χ.

Definition 2.11 (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

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

See Theorem 2.14 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]

We will need the the following Theorem:

Theorem 2.12 (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]⟩ ,

𝔉^{0} ≡ ∨_{(t,s)∈[0,1]^{2}}𝔉_{ts}^{0}, and χ is a 𝔎-valued Brownian bridge sheet in the sense of Definition 2.9. Let ∂_{t} denote Stratonowicz differentiation in the t variable. Then given k_{0} ∈ L (K) there is a jointly continuous solution Σ (t, s) to the stochastic differential equation

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

where the A run through an orthonormal basis of 𝔎 and χ^{A} (t, s) ≡ ⟨χ (t, s) , A⟩_{𝔎}. Henceforth we write Eq. (2.2) more concisely as

∂_{t}Σ (t, s) = (ℓ_{Σ(t,s)})_{∗} ∂_{t}χ (t, s) with Σ (0, s) = k_{0} (s) , ∀s ∈ [0, 1] . (2.3)

[see Malliavin [23]; see also Theorem 3.8 of [10] and Baxendale [6]]

Remark 2.13 (Explicit Matrix Representation of Eq. [2.3). ] 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. (2.3) can be rewritten as

(2.4)  ∂_{t}Σ (t, s) = Σ (t, s) ∂_{t}χ (t, s)
with Σ (0, ·) = k_{0}, ∀s ∈ [0, 1] ,

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

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

Theorem 2.14 (Brownian motion on L (K)). Let Σ (t, s) be the process from Theorem 2.12 and Remark 2.13. Theorem 2.12 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 2.11.

Proof. See Theorem 3.10 of Driver [10].

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

Definition 2.15 (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, ·) .

Remark 2.16 (Heat Kernel measure is a Heat Kernel). Driver and Lohrenz showed for any t > 0, for all bounded cylinder functions f on L (K); the function u on (0, ∞) × L (K) defined 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 2.2. See Theorem 1.1 of [12]. See also Definitions 3.10 and 4.17 in [12]. In [12], 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 2.17 (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 σ-algebra under P.

• Let Σ be the process from Theorem 2.12 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 2.14 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}.

Theorem 2.18 (Semimartingale properties of g_{·s}). The process g of Definition 2.17 has the following properties:-

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

2. Let X_{ts} ≡ ∫_{0}^{t} g_{τs}^{−1}∂_{τ} g_{τs}. 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.

Proof. t → Σ_{ts} a Brownian motion on K ⇒ t → g_{ts} a Brownian motion on K. In particular, g is a semimartingale and X_{ts} ≡ ∫_{0}^{t} g_{τs}^{−1}∂_{τ} g_{τs} is well-defined. By Proposition 8.3 of [9] we know that X_{ts} ∘ Σ = ∫_{0}^{t} Σ_{τs}^{−1}∂_{τ} Σ_{τs} = χ_{ts}. Thus X is a Brownian bridge sheet with respect to the measure P.

Remark 2.19. 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.

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

Theorem 2.20 (Semimartingale properties of g_{t·}). Let g be an L (K)-valued Brownian motion as in Defintion 2.17. Then:-

1. s → g_{ts} is a K-valued 𝔉_{ts}-semimartingale.

2.

∫_{0}^{s} ∂_{σ}g_{tσ}g_{tσ}^{−1} = W_{ts} − ∫_{0}^{s} \frac{dσ}{1 − σ} ∫_{0}^{t} Ad_{g_{τσ}}d_{τ} X_{τσ},

where s → W_{ts} is a Brownian motion on 𝔎 with parameter t.

Proof. Theorem 2.20 is a special case of Theorem 4.1 proved in Section 4.

Theorem 2.21. Let z < 1 and let 𝔊_{z} ≡ σ ⟨x_{s} : s ∈ [0, z]⟩ where x_{s} : L (K) → K is the evaluation map at time s. Then pinned Wiener measure, μ_{0}, is absolutely continuous with respect to Heat Kernel measure, ν_{1} (e, ·), on the σ-algebra 𝔊_{z}.

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

3. Motivation for Theorem 2.20

Definition 3.1 (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).

Remark 3.2. (Theorem 2.20 is reasonable)g satisfies

∂_{t}g_{ts} = g_{ts}∂_{t}X_{ts} with g_{0s} = e, (3.1)

where X_{·s} is the Brownian bridge sheet from Theorem 2.18. By Theorem 3.7, 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. (3.2)

If we replace X by b in Eq. (3.1), then Lemma 3.3 shows that s → g_{ts} would be a K-valued Brownian motion with variance t and hence ∫_{0}^{·} ∂_{s}g_{ts}g_{ts}^{−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 ∫_{0}^{·} ∂_{s}g_{ts}g_{ts}^{−1} is equivalent (but not equal) to the law of a Brownian motion on 𝔎.

Lemma 3.3 (Semimartingale properties of h_{t·}). Let b be a 𝔎-valued Brownian Sheet (see Definition 3.1)Let h_{ts} be the solution to

∂_{t}h_{ts} = h_{ts}∂_{t}b_{ts} with h_{0s} = e. (3.3)

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. [3.3] is to be interpreted like Eq. [2.2].

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

∂_{t} (h_{ts_{i}}h_{ts_{i−1}}^{−1})

= h_{ts_{i}}∂_{t}b_{ts_{i}}h_{ts_{i−1}}^{−1} − h_{ts_{i}}∂_{t}b_{ts_{i−1}}h_{ts_{i−1}}^{−1}

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

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

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

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

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

= (h_{ts_{i}}h_{ts_{i−1}}^{−1}) ∂_{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

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

It suffices to show that {∫_{0}^{·} Ad_{h_{ts_{i−1}}} d_{t}△_{i}b (t)}_{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}}} d_{τ} △_{i}b (τ) Ad_{h_{τs_{j−1}}} d_{τ} △_{j}b (τ) .

Then

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

= 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.

We still have to show that h_{ts} has a jointly continuous version. That is by Kolmogorov’s continuity criterion we must show that

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

where d (x, y) denotes the distance between points x and y in K. The proof is essentially the same as that done in Theorem 3.8 of Driver [10] with the modification that G (s, σ) is used in place of G_{0} (s, σ). in particular, see Eq. [3.12] of [10].

3.1. Semimartingale Properties of X_{ts}. Let X_{ts} be as in Theorem 2.18. 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 recall the decomposition of a Brownian bridge X̃ (below in Remark 3.4) in terms of a Brownian motion and a finite-variation part. The Brownian bridge X̃_{·} is supposed to play the role of X_{t·} but with one fewer parameter.

Remark 3.4 (Doob’s h transform). Let X̃ be a Brownian bridge from 0 to 0 on ℝ. Then there is a Brownian motion b̃ which can be written as a linear function of X̃

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

Definition 3.5. Define continuous 𝔎-valued linear maps on paths,

T_{s}, S_{s} : C ([0, 1] → 𝔎) → 𝔎,

by setting

T_{s}(y) = y(s) − ∫_{0}^{s} y(σ)\frac{(1 − s)}{(1 − σ)^{2}} dσ if s ∈ [0, 1) .
S_{s}(x) ≡ x(s) + ∫_{0}^{s} \frac{x(σ)}{(1 − σ)}dσ if s ∈ [0, 1) .

Remark 3.6. Notice that in Remark 3.4 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 3.7 (Decomposition of the Brownian bridge sheet). Let X be the Brownian bridge sheet from Theorem 2.18. 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. (3.4)

We shall defer the proof of Theorem 3.7 until after Lemma 3.8 below.

Lemma 3.8 (Properties of the transformations S and T). There exist unitary maps T : H (𝔎) → H_{0} (𝔎) and S : H_{0} (𝔎) → H (𝔎) so that T (y) (s) = T_{s} (y) and S (x) (s) = S_{s} (x) for any s ∈ [0, 1). Furthermore S = T^{−1}.

Proof. Define a subset of H (𝔎) by setting

U ≡ {y : y′ ∈ C_{c}^{∞} ((0, 1) → 𝔎)} .

C_{c}^{∞} ((0, 1) → 𝔎) is dense in L^{2} ([0, 1] → 𝔎) and therefore, by the isometry between L^{2} and H (𝔎),U is dense in H (𝔎).

Define x (s) = T_{s} (y) for y ∈ U. Then

x (s) = ∫_{0}^{s} \frac{(1 − s)}{1 − σ}y′(σ)dσ,

and

x′ (s) = y′ (s) − ∫_{0}^{s} \frac{y′(σ)}{1 − σ} dσ. (3.5)

Since y′ is zero near 1, s → y′(s)/ (1 − s) is bounded on [0, 1] and x′ is constant near s = 1. The boundedness of y′(s)/ (1 − s) implies that x (s) → 0 as s → 1. So we can define a map T : U → H_{0} (𝔎) .

We claim that T is a norm-preserving and so can be extended to a map from H (𝔎) onto a closed subspace of H_{0} (𝔎). Integrating by parts,

∫_{0}^{1}ds [∫_{0}^{s} \frac{y′(σ)}{1 − σ} dσ]^{2} − 2 ∫_{0}^{1} y′ (s) · ∫_{0}^{s} \frac{y′(σ)}{1 − σ} dσds

= [∫_{0}^{1} \frac{y′(σ)}{1 − σ} dσ]^{2} − 2 ∫_{0}^{1} \frac{y′(s)}{1 − s} · ∫_{0}^{s} \frac{y′(σ)}{1 − σ} dσds

= [∫_{0}^{1} \frac{y′(σ)}{1 − σ} dσ]^{2} − 2 ∫_{0}^{1} [∫_{0}^{s} \frac{y′(σ)}{1 − σ} dσ] · d_{s} [∫_{0}^{s} \frac{y′(σ)}{1 − σ} dσ]

= 0,

and so expanding with Eq. [3.5] we see that

∫_{0}^{1} |x′ (s)|^{2} ds = ∫_{0}^{1} |y′ (s)|^{2} ds.

If x ⊥ Im T, then for any y ∈ H (𝔎)

0 = ∫_{0}^{1} x′ (s) · [y′ (s) − ∫_{0}^{s} \frac{y′(σ)}{1 − σ} dσ] ds

⟺ ∫_{0}^{1} x′ (s) · y′ (s) ds = ∫_{0}^{1} (x (1) − x (σ)) · \frac{y′(σ)}{1 − σ} dσ

⟺ ∫_{0}^{1} [x′ (s) + \frac{x (s)}{1 − s}] · y′ (s) ds = 0.

Since y′ can be any arbitrary element of L^{2} ([0, 1] → 𝔎), we must have

x′ (s) = \frac{−x (s)}{1 − s} ⟺ x (s) = x (ε) \frac{1 − s}{1 − ε}.

Letting ε → 0 and using x (0) = 0, we see that x is identically 0. Thus T is surjective and so provides an isometry between H (𝔎) and H_{0} (𝔎).

Now we find the inverse of T. Let x in H (𝔎), y = T (x) in H_{0} (𝔎) and z_{s} = S_{s} (y) so that z_{s} = S_{s} ∘ T (x). Since x (0) = z_{0} = 0 and

\frac{d}{ds}z_{s} = y′ (s) + \frac{y (s)}{1 − s}
= x′ (s) − ∫_{0}^{s} \frac{x′ (σ) dσ}{1 − σ} + \frac{y (s)}{1 − s}
= x′ (s) − \frac{T (x) (s)}{1 − s} + \frac{y (s)}{1 − s}
= x′ (s) ,

we conclude that S_{s} ∘ T (x) = x (s) for any x in H (𝔎).

Define S : H_{0} (𝔎) → H (𝔎) so that S (y) (s) = S_{s} (y) for any s < 1 and S (y) (1) = lim_{s→1} S (y) (s). Then by definition, S is the inverse of T.

Proof. of Theorem 3.7:

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} d_{u}α_{s} (u) x (u) ,

where

d_{u}α_{s} (u) = [δ (u − s) + 1_{[0,s]} \frac{1}{1 − u}] du

is a positive measure on [0, 1]. Here δ denotes the Dirac delta measure. 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 3.5. So

Eb_{ts}^{A}b_{τσ}^{B} = E ∫ d_{u}α_{s} (u) d_{v}α_{σ} (ν) X_{tu}^{A}X_{τν}^{B}. (3.6)

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

E ∫ d_{u}α_{s} (u) d_{v}α_{σ} (ν) |X_{tu}^{A}X_{τν}^{B} |
≤ ∫ d_{u}α_{s} (u) d_{v}α_{σ} (ν) \sqrt{E (X_{tu}^{A})^{2} E (X_{τν}^{B})^{2}}
= ∫ d_{u}α_{s} (u) d_{v}α_{σ} (ν) \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} = ∫ d_{u}α_{s} (u) d_{v}α_{σ} (ν) EX_{tu}^{A}X_{τν}^{B}
(3.7)  = (t ∧ τ) ⟨A, B⟩_{𝔎} ∫ d_{u}α_{s} (u) ∫ d_{v}α_{σ} (ν) G_{0} (u, ν) .

Since G_{0} (u, ν) is the reproducing kernel for H_{0} (𝔎), we have for any orthonormal basis h of H_{0} (𝔎) ,

G_{0} (u, ν) = ∑ 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) d_{u}α_{s} (u) ∫ h (ν) d_{v}α_{σ} (ν)
= (t ∧ τ) ⟨A, B⟩_{𝔎} ∑ S_{s} (h) S_{σ} (h)
= (t ∧ τ) ⟨A, B⟩_{𝔎} ∑ S (h) (s) S (h) (σ) .

By Lemma 3.8, S (h) runs through an orthonormal basis of H (𝔎) . This, together with the fact that G (s, σ) is the reproducing kernel for H (𝔎) yields,

∑ S (h) (s) S (h) (σ) = G (s, σ) = s ∧ σ.

Thus

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

Thus b is a 𝔎-valued Brownian sheet.

It remains to show that T_{s} (b_{t·}) = X_{ts}. Let x ∈ L (𝔎). Then for any ε > 0, can choose a δ so that sup_{[0,δ]∪[1−δ,1]} |x| < ε. There is an x̃ on C^{∞} ((δ, 1 − δ) → 𝔎) so that ∥x̃ − x ↓_{(δ,1−δ)}∥_{∞} < ε. Define \overline{x} by setting

\overline{x} (s) = \frac{s}{δ}x̃ (δ) 1_{[0,δ]} + x̃ (s) 1_{(δ,1−δ)} + \frac{1 − s}{δ}x̃ (1 − δ) 1_{[1−δ,1]}.

Then

∥\overline{x} − x∥_{∞} < (2 sup_{[0,δ]∪[1−δ,1]} |x| + ε) ∨ ε < 3ε.

Furthermore, \overline{x} ∈ H_{0} (𝔎).

So now take x ∈ H_{0} (𝔎) so that ∥x − X_{t·}∥_{∞} < ε. Then T_{s} (b_{t·}) = T_{s} (b_{t·} − S (x))+ x (s).

|T_{s}(b_{t·} − S (x))| ≤ sup_{[0,s]} |b_{t·} − S (x)| (1 + ∫_{0}^{s} \frac{(1 − s)}{(1 − σ)^{2}} dσ) .

sup_{[0,s]} |b_{t·} − S (x)| ≤ sup_{u∈[0,s]} |X_{tu} − x (u) + ∫_{0}^{u} \frac{X_{tσ} − x (σ)}{(1 − σ)}dσ|

≤ sup_{u∈[0,s]} |X_{tu} − x (u)| (1 + ∫_{0}^{s} \frac{dσ}{(1 − σ)}) .

Thus as ε → 0 we have T_{s} (b_{t·}) = X_{ts} and we are done.

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

Let X be the Brownian bridge sheet from subsection 3.1. In this section we shall show that X is a semimartingale in the sense of Norris (see [24]) and then proceed to find ∫_{0}^{·} ∂_{s}g_{T s}g_{T s}^{−1} using his powerful two-parameter calculus methods. Throughout this section we shall use the term “semimartingale” to mean “semimartingale in the sense of Norris”. We shall also stick to Norris’ notation as far as possible in this section.

Theorem 4.1 (semimartingale properties of g_{T ·}). Let g be our L (K)-valued Brownian motion. Then:-

1. g is a semimartingale.

2. There is a 𝔎-valued Brownian sheet b̃ with the same law as b so that

∫_{0}^{s} ∂_{σ}g_{T σ}g_{T σ}^{−1} = b̃_{T s} − ∫_{0}^{s} \frac{dσ}{1 − σ} ∫_{0}^{T} Ad_{g_{tσ}}d_{t}X_{tσ}.

We defer the proof of this Theorem to the end of the section.

Theorem 4.2 (X is a semimartingale). On the domain 𝔇 ≡ {(t, s) ∈ [0, ∞) × [0, 1)}, X is a uniform semimartingale satisfying the equation

∂_{t}∂_{s}X = ∂_{t}∂_{s}b − \frac{1}{1 − s}∂_{t}X∂s with X_{0s} = X_{t0} = 0. (4.1)

Proof. First we show X is a uniform (s, t)-semimartingale in the sense of Norris. By Theorem 3.7 we have

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

Since b is clearly a semimartingale, it will suffice to show that the expression

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

is an (s, t)-semimartingale. Differentiating, we see that

C_{ts} = ∫_{0}^{s} dσ [\frac{b_{tσ}}{(1 − σ)} − ∫_{0}^{σ} b_{tu} \frac{1}{(1 − u)^{2}} du] . (4.2)

By Ito

b_{tσ} \frac{1}{(1 − σ)} = ∫_{0}^{σ} b_{tu} \frac{1}{(1 − u)^{2}} du + ∫_{0}^{σ} d_{u}b_{tu} \frac{1}{(1 − u)},

and so

C_{ts} = ∫_{0}^{s} dσ ∫_{0}^{σ} \frac{1}{(1 − u)}d_{u}b_{tu}
= ∫_{−1}^{t} ∫_{0}^{s} [∫_{0}^{t} ∫_{−1}^{s} \frac{1}{(1 − u′)}1_{[0,u]} (u′) 1_{[−1,0]} (r′) d_{r}d_{u′}b_{ru′}] dr′du.

We still need to check Eq. (2.15) of Norris; that for any p ∈ [1, ∞), the expression

∫_{−1}^{t} ∫_{0}^{s} [∫_{0}^{t} ∫_{−1}^{s} ∥\frac{1}{(1 − u′)}1_{[0,u]} (u′) 1_{[−1,0]} (r′)∥_{L^{p}}^{2} drdu′]^{1/2} dr′du

is finite. A direct computation establishes that this is so for any (t, s) ∈ 𝔇. Thus X is a uniform (s, t)-semimartingale. By Norris (see Pg 282 of [24]) we know that a uniform (s, t)-semimartingale which is constant on both the t and s axes is both a uniform t-semimartingale as well as a uniform s-semimartingale. Since X_{0s} = X_{t0} = 0, we conclude that X is a uniform semimartingale.

Now that we have shown X is a semimartingale, we can verify by direct computation that X satisfies Eq. [4.1].

∂_{t}∂_{s}X − ∂_{t}∂_{s}b + \frac{1}{1 − s}∂_{t}X∂s
= ∂_{t}X∂s − ∂_{t}∂_{s}C_{ts}.

¿From Eq. [4.2] we see that

∂_{t}∂_{s}X − ∂_{t}∂_{s}b + \frac{1}{1 − s}∂_{t}X∂s

= \frac{∂_{t}X∂s}{1 − s} − [∂_{t}b_{ts} − ∫_{0}^{s} ∂_{t}b_{tσ} \frac{1 − s}{(1 − σ)^{2}} dσ] \frac{∂s}{1 − s}

= 0.

Hence we are done.

Lemma 4.3 (g is a semimartingale). Let g be our L (K)-valued Brownian motion from Definition 2.17. Then on 𝔇, g is a semimartingale satisfying

D_{s}∂_{t}ψ = ψ∂_{t}∂_{s}b − \frac{1}{1 − s}∂_{t}ψ∂s with g_{0s} = g_{t0} = e.

Proof. Define ψ to be the solution of

D_{s}∂_{t}ψ = ψ∂_{t}∂_{s}b − \frac{1}{1 − s}∂_{t}ψ∂s, (4.3)

where D denotes covariant differentiation with respect to the left connection on K. Apply Theorem 3.2.6 of [24] to the ℝ × K-valued process ψ̃_{ts} ≡ (s, ψ_{ts}). To facilitate this define coordinate projections π_{1} : (s, x) → s and π_{2} : (s, x) → x. Then we have

D̃_{s}∂_{t}ψ̃ = (0, π_{2} \binom{}{ψ} ∂_{t}∂_{s}b) − \frac{1}{1 − s}∂_{t}ψ̃∂s,

where D̃ denotes covariant differentiation with respect to the left connection on ℝ × K. In Norris notation, we would have

D̃_{s}∂_{t}ψ̃ = α ⟨∂_{t}∂_{s}b⟩ + β_{01} ⟨∂_{t}ψ̃∂s⟩ ,

with

α ⟨ω⟩ = (0, π_{2} (x) ω) ∀ω ∈ 𝔎,

and

β_{01} ⟨ω⟩ = (\frac{−1}{1 − π_{1} (x)}) ω ∀ω ∈ T_{x} (ℝ × K) .

Thus ψ̃ is a semimartingale, and hence so is ψ.

Define

χ_{·s} = ∫_{0}^{·} ψ_{ts}^{−1}∂_{t}ψ_{ts}
= ∫_{0}^{·} ψ_{ts}^{−1}d_{t}ψ_{ts} − \frac{1}{2} ∫_{0}^{·} (ψ_{ts}^{−1}d_{t}ψ_{ts}) (ψ_{ts}^{−1}d_{t}ψ_{ts})
= I − J.

The first term I is a semimartingale by Theorem 2.3.1 of [24], while the second term J is a semimartingale by Theorem 2.3.2 of [24]. Thus χ is a semimartingale. We will now show that χ satisfies Eq. [4.1]. Computing directly, we see that

∂_{t}∂_{s}χ + \frac{1}{1 − s}∂_{t}χ∂s

= ∂_{s} (ψ^{−1}∂_{t}ψ) + \frac{1}{1 − s}ψ^{−1}∂_{t}ψ∂s

= ψ^{−1}D_{s}∂_{t}ψ + \frac{1}{1 − s}ψ^{−1}∂_{t}ψ∂s.

Applying Eq. [4.3], we see that

∂_{t}∂_{s}χ + \frac{1}{1 − s}∂_{t}χ∂s = ∂_{t}∂_{s}b.

Thus χ = X and ψ = g. Thus g is a semimartingale satisfying the equation

D_{s}∂_{t}g = g∂_{t}∂_{s}b − \frac{1}{1 − s}∂_{t}g∂s.

We are now able to return to the proof of Theorem 4.1.

Proof of Theorem 4.1. We have already shown that g is a semimartingale. Let D and D̂ denote covariant differentiation with respect to the left and right connections on K respectively. Thus

D̂_{t}∂_{s}g = (∂_{t} ((∂_{s}g) g^{−1})) g
= ∂_{t}∂_{s}g − (∂_{s}g) g^{−1} (∂_{t}g)
= g∂_{s} (g^{−1}∂_{t}g)
= D_{s}∂_{t}g
= g∂_{t}∂_{s}b − \frac{1}{1 − s}∂_{t}g∂s.

Now define a process b̃ by setting

∂_{t}∂_{s}b̃ = Ad_{g} ⟨∂_{t}∂_{s}b⟩ with b̃_{0s} = b̃_{t0} = 0.

Ad_{g} is previsible and preserves the inner product on 𝔎 and so by Theorem 2.4.1 of [24], we see that b̃ is a Brownian sheet on 𝔎 (i.e. b̃ and b have the same distribution). Therefore,

(∂_{t} ((∂_{s}g) g^{−1})) g = D̂_{t}∂_{s}g = (∂_{t}∂_{s}b̃) g − ∂_{t}g \frac{ds}{1 − s},

which implies that

∂_{t} ((∂_{s}g) g^{−1}) = (∂_{t}∂_{s}b̃) − Ad_{g} ⟨∂_{t}X⟩ \frac{ds}{1 − s}.

This means that

∫_{0} (∂_{s}g) g^{−1} = b̃ − ∫_{0} ∫_{0} Ad_{g} ⟨∂_{t}X⟩ \frac{ds}{1 − s}.

Now using the fact that d_{t}g = gd_{t}X + \frac{1}{2}gd_{t}Xd_{t}X, we see that

Ad_{g}∂_{t}X = Ad_{g}d_{t}X

+\frac{1}{2} (d_{t}gd_{t}X) g^{−1} − \frac{1}{2}g (d_{t}X) g^{−1} (d_{t}g) g_{ts}^{−1}

= Ad_{g}d_{t}X.

Thus we have shown that

∫_{0} (∂_{s}g) g^{−1} = b̃ − ∫_{0} \frac{ds}{1 − s} ∫_{0} Ad_{g} ⟨d_{t}X⟩ .

5. HKM ↓_{𝔊_{z}}∽ P W M ↓_{𝔊_{z}}

Let Ω, P, and g_{t} be as in Definition 2.17 and let 𝔊_{z} be as in Theorem 2.21. Then t → g_{t} is an L (K)-valued Brownian motion and thus Law g_{t} equals Heat Kernel measure ν_{t} (e, ·). From Section 4 we know that ∫_{0}^{s} ∂_{σ}g_{tσ}g_{tσ}^{−1} is a Brownian semimartingale. In this Section we will show that ∫_{0}^{s} ∂_{σ}g_{tσ}g_{tσ}^{−1} has a law equivalent to that of a Brownian motion. We will then know that Pinned Wiener measure is equivalent to Heat Kernel measure on 𝔊_{z} for any z < 1, by the equivalence of Wiener measure and Pinned Wiener measure.

Theorem 5.1. 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 after the statement of Lemma 5.4.

Definition 5.2. Let B_{ts} be defined to solve the Fisk-Stratonowicz equation ∂_{s}B_{ts} = ∂_{s}b_{ts}B_{ts} with B_{t0} = e where b is the Brownian sheet from Theorem 3.7.

Theorem 5.3. Let t ↦ g_{t·} be our L (K)-valued Brownian motion from Definition 2.17 and let s → B_{ts} be the K-valued Brownian motion of Definition 5.2. 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 in section 5.1.

Lemma 5.4. If κ_{1} ∽ κ_{2} then κ_{1} ⊗ ν ∽ κ_{2} ⊗ ν, where κ_{1}, κ, ν are probability measures.

We prove this Lemma after the proof of Theorem 5.1 below.

Lemma 5.5. Let 0 < t_{1} < ⋯ < t_{k}. Then B_{t_{i−1}·}^{−1}B_{t_{i}·} are independent 𝔎-valued Brownian motions with parameters t_{i} − t_{i−1}.

The proof of this Lemma is supplied after the proof of Lemma 5.4

Proof of Theorem 5.1. Let μ_{0} be Pinned Wiener measure on L (K) and let μ be Wiener measure on C ([0, 1] → K) as in Definitions 2.5 and 2.7. Then μ = Law [B_{1·}] since B_{1·} is a standard K-valued Brownian motion by Definition 5.2. 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. We know that t → g_{t·} is an L (K)-valued Brownian motion and so has independent increments. Suppose we can show for t_{2} > t_{1} that B_{t_{1}s}^{−1}B_{t_{2}s} is independent of 𝔉_{t_{1}1}. Then letting A′ = {(k_{1}, ⋯, k_{n}) : k_{1} ⋯ k_{N} ∈ A} we have

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 ·}}) (A′) ,

while

μ (A) = P {B_{1·} ∈ A}
= P {B_{(1/N)·} (B_{(1/N)·}^{−1}B_{(2/N)·}) ⋯ (B_{(N−1/N)·}^{−1}B_{1·}) ∈ A}
= (⊗_{i=1}^{N}Law_{B_{T ·}}) (A′) .

Now by Theorem 5.3 g_{T ·} has a law equivalent to that of B_{T ·}, on the restricted σalgebra 𝔊_{z}. Invoking Lemma 5.4 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, A′ ∈ 𝔊_{z}^{⊗N} and

ν_{1} (e, A) = (⊗_{i=1}^{N}Law_{g_{T ·}}) (A′) . = 0
⟺ μ (A) = (⊗_{i=1}^{N}Law_{B_{T ·}}) (A′) = 0.

Hence we are done if we show that B_{t_{1}s}^{−1}B_{t_{2}s} is independent of 𝔉_{t_{1}1}.

By Lemma 5.5, if τ ≤ t_{1} < t_{2} then B_{τ·} and B_{t_{1}·}^{−1}B_{t_{2}·} are independent Brownian motions and so B_{t_{1}·}^{−1}B_{t_{2}·} is independent of σ ⟨B_{ts} : t ≤ t_{1} and s ≤ 1⟩ . However since b_{ts} = ∫_{0}^{s} B_{tδu}B_{tu}^{−1}, we see that

𝔉_{t_{1}1} = σ ⟨b_{ts} : t ≤ t_{1} and s ≤ 1⟩
⊂ σ ⟨B_{ts} : t ≤ t_{1} and s ≤ 1⟩ .

Therefore B_{t_{1}·}^{−1}B_{t_{2}·} is independent of 𝔉_{t_{1}1} and we are finished.

Proof of Lemma 5.4. It 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.

Proof of Lemma 5.5. ¿From Definition 5.2 we see that

∂_{s} (B_{t_{i−1}s}^{−1}B_{t_{i}s}) = [Ad_{B_{t_{i−1}s}^{−1}}∂_{s} (b_{t_{i}s} − b_{t_{i−1}s})] B_{t_{i−1}s}^{−1}B_{t_{i}s}. (5.1)

Let b̃_{i} ≡ ∫_{0}^{·} Ad_{B_{t_{i−1}s}^{−1}}∂_{s} (b_{t_{i}s} − b_{t_{i−1}s}) . Then

b̃_{i} = ∫_{0}^{·} Ad_{B_{t_{i−1}s}^{−1}}d_{s} (b_{t_{i}s} − b_{t_{i−1}s}) + \frac{1}{2} ∫_{0}^{·} Ad_{B_{t_{i−1}s}^{−1}} [d_{s} (b_{t_{i}s} − b_{t_{i−1}s}) , d_{s}b_{t_{i−1}s}]
= ∫_{0}^{·} Ad_{B_{t_{i−1}s}^{−1}}d_{s} (b_{t_{i}s} − b_{t_{i−1}s}) .

Thus since the Adjoint action is norm preserving, we have that b̃_{i} is a Brownian motion on 𝔎. Computing quadratic variations, we see that

d_{s}b̃_{i} (s) ⊗ d_{s}b̃_{j} (s)

= Ad_{B_{t_{i−1}s}^{−1}}d_{s} (b_{t_{i}s} − b_{t_{i−1}s}) ⊗ Ad_{B_{t_{j−1}s}^{−1}}d_{s} (b_{t_{j}s} − b_{t_{j−1}s})

= ∑_{A} (Ad_{B_{t_{i−1}s}^{−1}}A ⊗ Ad_{B_{t_{j−1}s}^{−1}}A) d_{s} (b_{t_{j}s}^{A} − b_{t_{j−1}s}^{A}) d_{s} (b_{t_{i}s}^{A} − b_{t_{i−1}s}^{A})

= δ_{ij} (∑_{A} A^{⊗2}) (t_{i} − t_{i−1}) ds.

5.1. Proof of Theorem 5.3.

Theorem 5.6. Let Y_{T s} be defined to be ∫_{0}^{s} ∂_{σ}g_{T σ}g_{T σ}^{−1} as in Theorem 4.1. Then Law Y_{T ·} ∽ Law b_{T ·} as measures on C ([0, s] → 𝔎) for any s < \frac{\sqrt{1+4T} −1}{2T}.

The proof of Theorem 5.6 is given in section 5.1.1

Proof of Theorem 5.3. Fix s. Pick T so that s < \frac{\sqrt{1+4T} −1}{2T}.

Let x_{σ} be the evaluation map at σ on C ([0, s] → 𝔎). Define the probability spaces Ω^{b} (resp. Ω^{Y}) as the set C ([0, s] → 𝔎) as equipped with Law (b_{T ·} ↓_{[0,s]}) (resp. Law (Y_{T ·} ↓_{[0,s]})) and filtration generated by x. Let η^{b} and η^{Y} be the solutions to the stochastic differential equation

∂η = η∂x with η_{0} = 1 (*)

in the probability spaces Ω^{b} and Ω^{y} respectively. Then Proposition 8.3 of [9] implies that η^{b} (b_{T ·} ↓_{[0,s]}) = B_{T ·} ↓_{[0,s]} and η^{Y} (Y_{T ·} ↓_{[0,s]}) = g_{T ·} ↓_{[0,s]}. Theorem 5.6 implies that b_{T ·} ↓_{[0,s]} and Y_{T ·} ↓_{[0,s]} have equivalent laws and hence η ≡ η^{b} = η^{Y} a.s. Thus if

E1_{A} (g_{T ·} ↓_{[0,s]}) = 0
⟺ E1_{A} ∘ η (Y_{T ·} ↓_{[0,s]}) = 0
⟺ E1_{A} ∘ η (b_{T ·} ↓_{[0,s]}) = 0
⟺ E1_{A} (B_{T ·} ↓_{[0,s]}) = 0.

Hence by the Radon-Nikodym Theorem Law g_{T ·} is equivalent to Law B_{T ·} as measures on C ([0, s] → K).

5.1.1. Proof of Theorem 5.6.

Remark 5.7. 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.

Remark 5.8 (Theorem 5.6 is not obvious). 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}}d_{t}d_{σ}X_{tσ}
∽ Law ∫_{R_{T ·}} Ad_{g_{tσ}}d_{t}d_{σ}b_{tσ} = 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.

Theorem 5.9 (Girsanov, see [19]). 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

Z̃_{·} ≡ exp [∫_{0}^{·} Z_{s} · d_{s}β_{s} − \frac{1}{2} ∫_{0}^{·} |Z_{s}|^{2} ds] .

Define a new measure P̃_{S} on 𝔉_{S} by setting P̃ (A) = E1_{A}Z̃_{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.

We will use the following two Lemmas which are proven in section 5.1.2.

Lemma 5.10. Let X be the 𝔎-valued Brownian bridge sheet of Theorem 2.18. Then the expression X̃_{tσ} ≡ ∫_{0}^{t} Ad_{g_{uσ}} d_{u}X_{uσ} has the same law as X_{tσ}.

Lemma 5.11. Let X be the 𝔎-valued Brownian bridge sheet of Theorem 2.18. Then

P exp [\frac{1}{2} ∫_{0}^{s} dσ |∫_{0}^{T} Ad_{g_{tσ}} \frac{d_{t}X_{tσ}}{1 − σ} |_{𝔎}^{2}] < ∞, if s < \frac{\sqrt{1 + 4T} − 1}{2T}.

Proof of Theorem 5.6. Define

Z_{T} (σ) ≡ \frac{−1}{(1 − σ)} ∫_{0}^{T} Ad_{g_{tσ}} d_{t}X_{tσ}.

By definition of Y_{T ·} in Theorem 4.1

Y_{T ·} ≡ b̃_{T ·} + ∫_{0}^{·} dσZ_{T} (σ) .

By Lemma 5.11,

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) · d_{s}b̃_{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} ∘ b̃_{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.]

5.1.2. Proofs of Lemmas 5.10 and 5.11.

Proof of Lemma 5.10. 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

d_{t}X̃_{tσ} ⊗ d_{t}X̃_{tσ} = Ad_{g_{tσ}} d_{t}X_{tσ} ⊗ Ad_{g_{tσ}} d_{t}X_{tσ}
= (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.

Proof of Lemma 5.11. By using Jensen’s inequality on the probability 1_{[0,s]} \frac{dσ}{s} we have

exp [\frac{1}{2} ∫_{0}^{s} dσ |∫_{0}^{T} Ad_{g_{tσ}} \frac{d_{t}X_{tσ}}{1 − σ} |_{𝔎}^{2}]

= exp [∫_{0}^{s} \frac{dσ}{s} \frac{s}{2} |∫_{0}^{T} Ad_{g_{tσ}} \frac{d_{t}X_{tσ}}{1 − σ} |_{𝔎}^{2}]

≤ ∫_{0}^{s} \frac{dσ}{s} exp [\frac{s}{2} |∫_{0}^{T} Ad_{g_{tσ}} \frac{d_{t}X_{tσ}}{1 − σ} |_{𝔎}^{2}] .

Thus

J ≡ P exp [\frac{1}{2} ∫_{0}^{s} dσ |∫_{0}^{T} Ad_{g_{tσ}} \frac{d_{t}X_{tσ}}{1 − σ} |_{𝔎}^{2}]

≤ ∫_{0}^{s} \frac{dσ}{s} P exp [\frac{s}{2 (1 − σ)^{2}} |∫_{0}^{T} Ad_{g_{tσ}}d_{t}X_{tσ}|_{𝔎}^{2}] .

Now by Lemma 5.10, X̃_{·} = ∫_{0}^{·} Ad_{g_{tσ}}d_{t}X_{tσ} is a Brownian motion on 𝔎 with pa-

rameter G_{0} (σ, σ). Thus in the expectation, we can replace |∫_{0}^{T} Ad_{g_{tσ}}d_{t}X_{tσ}|_{𝔎}^{2} by

T G_{0} (σ, σ) |N|_{𝔎}^{2}, where N is a standard normal 𝔎-valued random variable. So we get

J ≤ ∫_{0}^{s} \frac{dσ}{s} P exp \frac{sT σ}{2 (1 − σ)} |N|_{𝔎}^{2} .

So we see that

J < ∞ ⟺ \frac{sT σ}{(1 − σ)} < ∞ 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}) .

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