ABSOLUTE CONTINUITY OF HEAT KERNEL MEASURE WITH PINNED WIENER MEASURE ON LOOP GROUPS

BRUCE K. DRIVER^{†} AND VIKRAM K. SRIMURTHY^{∗}

Abstract. Let t > 0, K be a connected compact Lie group equipped with an Ad_{K}-invariant inner product on the Lie Algebra of K. Associated to this data are two measures μ_{t}^{0} and ν_{t}^{0} on L(K) – the space of continuous loops based at e ∈ K. The measure μ_{t}^{0} is pinned Wiener measure with “variance t” while the measure ν_{t}^{0} is a “heat kernel measure” on L(K). The measure μ_{t}^{0} is constructed using a K – valued Brownian motion while the measure ν_{t}^{0} is constructed using a L(K) – valued Brownian motion. In this paper we show that ν_{t}^{0} is absolutely continuous with respect to μ_{t}^{0} and the Radon-Nikodym derivative dν_{t}^{0}/dμ_{t}^{0} is bounded.

Contents

1. Introduction 2 1.1. Conjecture on equivalence 4 2. Notation and Statements of Results 4 2.1. Brownian Sheets 4 2.2. K– valued Brownian motion and Wiener measures 6 2.3. Heat kernel measure on W (K) and L(K) 10 2.4. Statement of Results 11 3. Generators of Σ(t, ·) and Σ^{0}(t, ·) 12 3.1. Cameron-Martin spaces 12 3.2. Derivatives and Laplacians on L(K) and W (K) 14 3.3. Heat equations 15 4. The path group case 17 4.1. Proof of Theorem 2.15 18 5. Proof of the Airault-Malliavin Theorem 2.18 21 5.1. Integration by parts and strong differentiability 21 5.2. Proof of Theorem 2.18 25 6. Absolute continuity of heat kernel with respect to pinned Wiener measure 28 6.1. Proof of Theorem 2.16 30 7. The K = ℝ^{d} and S^{1} cases 32 7.1. The K = ℝ^{d} case 33 7.2. The K = S^{1} case 33 8. Appendix (Quadratic variations) 36 References 37

Date: April 14, 2000 File:redfinal.tex. 1991 Mathematics Subject Classification. Primary: 60H07, 58D30 Secondary 58D20. Key words and phrases. Loop groups, heat kernel measures, absolute continuity. †This research was partially supported by NSF Grants DMS 96-12651 and DMS 99-71036.

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

1. Introduction

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. To simplify notation later we will assume that K is a matrix group. (Since K is compact, this is no restriction, see for example Theorem 4.1 on p. 136 in [7].)

Example 1.1. As an example, let K = SO(3) be the group of 3×3 real orthogonal matrices with determinant 1. The Lie algebra of K is 𝔨 = so(3), the set of 3 × 3 real skew symmetric matrices, and the inner product ⟨A, B⟩_{𝔨} := −tr(AB) is an example of an Ad_{K} – invariant inner product on 𝔨.

Elements A ∈ 𝔨 will be identified with the unique left invariant vector field on K agreeing with A at the identity in K, i.e. if f ∈ C^{∞}(K) then

Af(x) = \frac{d}{dt}|_{0}f(xe^{tA}).

The path and loop groups on K are defined by

W (K) ≡ {σ ∈ C ([0, 1] → K) |σ (0) = e} (1.1)

and

L (K) ≡ {σ ∈ W (K) |σ (1) = e} (1.2)

respectively.

Notation 1.2. The constant path at e will be denote by e, i.e. e(s) = e for s ∈ [0, 1].

Pinned Wiener measure (μ_{t}^{0}) on such a “loop group” (see [21], [23],[3], [17] and Definition 2.11 below) is the law of a K-valued Brownian motion starting at e ∈ K and conditioned to end at e ∈ K. Heat kernel measure (ν_{t}^{0}) on L(K) (see [18], [13], and Carson [8, 9] and Definition 2.14 below) is the end point distribution of a “L(K) – valued Brownian motion.” The main theorem (Theorem 2.16) in this paper asserts that ν_{t}^{0} is absolutely continuous with respect to μ_{t}^{0} and the Radon-Nikodym derivative dν_{t}^{0}/dμ_{t}^{0} is bounded. The proof of this theorem heavily relies on a theorem of Airault and Malliavin (Theorem 2.18 below) which shows that μ_{t}^{0} solves a heat equation with a potential. A new proof of Theorem 2.18 will be given in Section 6.

One of our motivations for investigating Theorem 2.16 is L. Gross’ logarithmic Sobolev inequality on (L(K), μ_{t}^{0}). To state the inequality, let

∥gradf∥^{2} = ∑_{h∈S_{0}} (∂_{h}f)^{2} ,

where S_{0} is an orthonormal basis for H_{0} (H_{0} is the 𝔨 – valued Cameron -Martin space in Definitions 3.1) and ∂_{h} is a left invariant vector field on L(K) defined in Definition 3.4. Also let us introduce the following notation. If μ is a measure on some measurable space Ω and f : Ω → ℝ is a measurable function, let

μ(f) = ∫_{Ω} fdμ. (1.3)

L. Gross proves in [20] that there is a constant C < ∞ such that

∫_{L(K)} f^{2} log \frac{f^{2}}{μ_{t}^{0}(f^{2})}dμ_{t}^{0} ≤ C ∫_{L(K)} \{∥gradf∥^{2} + V f^{2}\} dμ_{t}^{0} (1.4)

where V is essentially the same potential that appears in the Airault – Malliavin Theorem 2.18 below. It is still an open question as to whether the potential term V f^{2} in Eq. (1.4) is necessary or not.

On the other hand, it was shown in Driver and Lohrenz [18] that if μ_{t}^{0} is replaced by ν_{t}^{0}, the potential term V is not needed, i.e. there is a constant C < ∞ such that

∫_{L(K)} f^{2} log \frac{f^{2}}{ν_{t}^{0}(f^{2})}dν_{t}^{0} ≤ C ∫_{L(K)} ∥gradf∥^{2} dν_{t}^{0}. (1.5)

Now Theorem 2.16 below shows that Z_{t} := dν_{t}^{0}/dμ_{t}^{0} is bounded. If one could show that Z_{t}^{−1} were also bounded, then the Holley – Stroock lemma (see [22] and Remark 1.20 in [10]) along with Eq. (1.5) would imply that Eq. (1.4) holds without the V f^{2} term. It is almost certainly seems too much to expect that Z_{t} is bounded from below in general. (It is not even known if Z_{t} > 0, μ_{t}^{0} - a.s., when K is non-abelian.) So the authors do not expect this line of reasoning to work without modification. Nevertheless, better knowledge of the density Z_{t} may be useful in determining if potential is needed in Eq. (1.4).

1.1. Conjecture on equivalence. Let us end this introduction with the following conjecture. Conjecture. If K is simply connected (so that L(K) has only one connected component) then Z_{t} > 0, μ_{t}^{0} - a.s. That is to say μ_{t}^{0} is absolutely continuous relative to ν_{t}^{0}. If K is not simply connected, then we expect that μ_{t}^{0} is absolutely continuous relative to a sum of left translates of ν_{t}^{0} by finite energy loops from each homotopy class.

The explicit calculations in Section 7 shows that the conjecture is true for K = ℝ^{d} and K = S^{1}, see Lemma 7.1 and Proposition 7.5. Moreover, the results in Srimurthy [31] also support the conjecture. Let F_{α} be the σ – algebra consisting of the measurable sets in W (K) depending only on the portion of the paths in W (K) over the interval [0, α], see Definition 2.5 below. Srimurthy proves that μ_{t}^{0} and ν_{t}^{0} are equivalent on F_{α} for any α < 1. Of course these σ – algebras are not able to detect the homotopy classes in L(K) and it is certainly not true that μ_{t}^{0} is absolutely continuous with respect to ν_{t}^{0} if K is not simply connected. This is because pinned Wiener measure μ_{t}^{0} charges all of the homotopy classes of K while the heat kernel measure ν_{t}^{0} only charges the trivial homotopy class.

2. Notation and Statements of Results

2.1. Brownian Sheets.

Definition 2.1 (𝔎 – valued Brownian Sheet). Let {β(t, s)}_{0≤s≤1,0≤t<∞} be a 𝔨 – valued Brownian sheet and {χ(t, s)}_{0≤s≤1,0≤t<∞} be a 𝔨 – valued Brownian bridge sheet defined on some probability space (Ω, G, P) . To be more precise, let s ∧ σ ≡ min(s, σ), G_{0} (s, σ) = s ∧ σ − σs, β^{A}(t, s) = ⟨A, β(t, s)⟩ and χ^{A}(t, s) = ⟨A, χ(t, s)⟩. Then we are assuming the β and χ are centered Gaussian random fields with covariance functions

𝔼[β^{A}(t, s)β^{B}(τ, σ)] = ⟨A, B⟩(t ∧ τ)(s ∧ σ) (2.1)

for all s, σ, t, τ ∈ [0, ∞) and A, B ∈ 𝔨 and

𝔼[χ^{A}(t, s)χ^{B}(τ, σ)] = ⟨A, B⟩(t ∧ τ)G_{0} (s, σ) (2.2) for all s, σ ∈ [0, 1], t, τ ∈ [0, ∞) and A, B ∈ 𝔨. (Here and in the sequel we will use 𝔼 to denote the expectation relative to the measure P.)

It is well known that β(t, s) and χ(t, s) may be chosen to have continuous sample paths, see for example the discussion after the proof of Corollary 1.3 in [33]. This fact may also be proved by abstract Wiener space considerations, see Remark 3.3 in [14]. So in the sequel we will assume that (t, s) → β(t, s) and (t, s) → χ(t, s) are continuous processes.

Definition 2.2. A 𝔨 – valued process {B_{s}} is said to be a Brownian motion with variance t if \frac{1}{\sqrt{t}}B_{s} is a standard 𝔨 – valued Brownian motion. Alternatively, B may be described using Lévy’s characterization (see for example Theorem 39 on p.80 in [27]) of Brownian motion, by requiring {B_{s}} to be a mean zero martingale with quadratic co-variations given by dB_{s}^{C}dB_{s}^{D} = t⟨C, D⟩ds for all C, D ∈ 𝔨.

Remark 2.3. Notice that for fixed s; t → β(t, s) and t → χ(t, s) are 𝔨 – valued Brownian motions with variance s and G_{0}(s, s) respectively. This follows by the independent increments of these processes in the t variable, Lemma 8.1 of the Appendix, and Definition 2.2. Similarly for fixed t; s → β(s, t) is a 𝔨 – valued Brownian motion with variance t. The process s → χ(t, s) is a Brownian Bridge for 0 ≤ s ≤ 1 with quadratic co-variation given by χ^{A}(t, ds)χ^{B}(t, ds) = t⟨A, B⟩ds, see Remark 2.12 below.

Definition 2.4 (Cylinder Functions). For 0 ≤ s ≤ 1, let π_{s} : W (K) → K be the projection map π_{s}(σ) = σ(s). More generally if

ℙ = {0 = s_{0} < s_{1} < s_{2} < ⋯ < s_{n} < 1} (2.3)

is a partition of [0, 1], let s_{n+1} = 1 by convention and let π_{ℙ} : W (K) → K^{n} be given by

π_{ℙ}(σ) = (σ(s_{1}), σ(s_{2}), …, σ(s_{n})). (2.4)

A cylinder function f on W (K) or L(K) is a function of the form f = F ∘ π_{ℙ} for some partition ℙ and some measurable function F : K^{n} → ℝ. The function f is said to be bounded (smooth) provided that F is bounded (smooth).

Definition 2.5. For s ∈ [0, 1], let F_{s} denote the σ – algebra on W (K) generated by the smooth cylinder functions of the form f = F ∘ π_{ℙ} where ℙ runs through partitions as in Eq. (2.3) with s_{n} ≤ s. We will write F for F_{1}.

The σ – algebra, F, is the same as the Borel σ – algebra on W (K), where W (K) is equipped with topology of uniform convergence relative to a metric on K derived from a Riemannian metric on T K.

Remark 2.6. i) For notational simplicity when working on L(K), we have defined π_{ℙ} as in Eq. (2.4) rather than by π_{ℙ}(σ) = (σ(s_{1}), σ(s_{2}), …, σ(s_{n}), σ(s_{n+1})) which would be more natural on W (K). This results in a slightly smaller class of cylinder functions, but this is of no significance for our purposes.

The next result is well known, but we include it for the reader’s convenience.

Lemma 2.7. Suppose that Q is a finite measure on (W (K), F) and 1 ≤ p < ∞. Then the smooth cylinder functions are dense in L^{p}(W (K), F, Q).

Proof. Let M denote the smooth cylinder functions and H denote those functions in the L^{p}(W (K), F, Q) – closure of M which are also bounded. Then H is a vector space containing the constant functions and which clearly satisfies the property; if {f_{n}}_{n=1}^{∞} is a sequence of functions in H such that 0 ≤ f_{1} ≤ f_{2} ≤ f_{3} ≤ …, and f := lim_{n→∞} f_{n} is bounded, then f ∈ H. Since M is closed under multiplication, we may apply the monotone class theorem (see Theorem 8 on p. 7 in [27]) to conclude H contains all bounded F = σ(M) – measurable functions. Since (by the dominated convergence theorem) H is dense in L^{p}(W (K), F, Q), we are done.

2.2. K– valued Brownian motion and Wiener measures.

Definition 2.8 (Wiener Measure on W (K)). Fix t > 0, let {g_{s}}_{s∈[0,1]} denote the solution to the stochastic differential equation

dg_{s} = g_{s}β(t, δs) with g_{0} = e ∈ K, (2.5)

where β(t, δs) denotes the Stratonovich differential of the Brownian motion s → β(t, s). The Wiener measure with variance t on F is μ_{t} := Law(g_{·}).

Let 𝔨_{0} ⊂ 𝔨 be an orthonormal basis for 𝔨, and Δ_{K} be the second order elliptic operator,

Δ_{K} = ∑_{A∈𝔨_{0}} A^{2}. (2.6)

Since K is compact and hence uni-modular, Δ_{K} is the Laplace Beltrami operator for the left invariant Riemannian metric on K determined by ⟨·, ·⟩ on 𝔨 =T_{e}K, see for example Remark 2.2 in [15]. Using Itô’s lemma, one easily shows that {g_{s}}_{s∈[0,1]} is a diffusion process on K with generator \frac{1}{2}tΔ_{K}. Such a K – valued process will be called a Brownian motion on K with variance t.

Definition 2.9 (Heat Kernel on K). Let p_{t}^{K} denote the smooth function of K such that Law(g_{1}) = p_{t}^{K}(x)dx, where dx denotes normalized Haar measure on K.

The function p_{t}^{K} is the convolution kernel for the heat operator e^{tΔ_{K}/2}. In particular, (t, x) → p_{t}^{K}(x) is a smooth positive function such that for any f ∈ C(K), the function u defined by

u (t, x) ≡ ∫_{K} f (y) p_{t}^{K} (x^{−1}y) dy for (t, x) ∈ (0, ∞) × K

satisfies the heat equation

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

where ∂_{t} = ∂/∂t.

Remark 2.10. It is well known that p_{t}^{K} (x) = p_{t}^{K} (x^{−1}) for all x ∈ K, see for example Item 2 of Proposition 3.1 in [15]. It also well known that p_{t}^{K} is a class function, i.e.

p_{t}^{K} (xy) = p_{t}^{K} (yx) for all x, y ∈ K. (2.7)

This is a consequence of the fact that Δ_{K} is a bi-invariant differential operator because of the Ad_{K} – invariance of ⟨·, ·⟩. Thus for all bounded measurable functions f on K,

∫_{K} f(y)p_{t}^{K} (x^{−1}y) dy = ∫_{K} f(xy)p_{t}^{K} (y) dy
= (e^{tΔ_{K}/2}f ∘ L_{x}) (e) = (e^{tΔ_{K}/2}f) (x) = (e^{tΔ_{K}/2}f ∘ R_{x}) (e)
= ∫_{K} f(yx)p_{t}^{K} (y) dy = ∫_{K} f(y)p_{t}^{K} (yx^{−1}) dy,

where L_{x} and R_{x} denote left and right multiplication by x ∈ K respectively. The last displayed equation implies Eq. (2.7).

By the Markov property of g_{·} and the previous comments, if f is a bounded cylinder function of the form f (σ) = F ∘ π_{ℙ} where ℙ is as in Eq. (2.3), then

μ_{t}(f) ≡ ∫_{K^{n}} F (x_{1}, ⋯, x_{n}) ∏_{i=1}^{n} p_{tΔ_{i}s}^{K} (x_{i−1}^{−1}x_{i}) dx_{1}dx_{2} … dx_{n}, (2.8)

where x_{0} := e and Δ_{i}s = s_{i} − s_{i−1}.

Definition 2.11 (Doob’s Construction of Pinned Wiener Measure). Pinned Wiener measure, μ_{t}^{0}, on W (K) with variance t, is the unique measure on F such that if f is a bounded F_{α} measurable function for some α ∈ (0, 1), then

μ_{t}^{0}(f) ≡ \frac{1}{p_{t}^{K}(e)}μ_{t}(fp_{t(1−α)}^{K}(π_{α})).

In particular if f is a bounded cylinder function f of the form f (σ) = F ∘ π_{ℙ} where ℙ is as in Eq. (2.3), then

μ_{t}^{0}(f) ≡ ∫_{K^{n}} F (x)ρ^{ℙ}(t, x)dx, (2.9)

where x = (x_{1}, ⋯, x_{n}) , dx = dx_{1}dx_{2} … dx_{n} is normalized Haar measure on K^{n} and

ρ^{ℙ}(t, x) := \frac{1}{p_{t}^{K}(e)} ∏_{i=1}^{n+1} p_{t(s_{i}−s_{i−1})}^{K} (x_{i−1}^{−1}x_{i}) (2.10)

where by convention x_{0} = x_{n+1} = e.

The existence of the probability measure μ_{t}^{0} and the fact that μ_{t}^{0}(L(K)) = 1 is well known. A proof may be found, for example, in Theorem 2.3 in [11]. To apply this theorem, the reader should take the covariant derivative ∇ appearing in Theorem 2.3 in [11] to be the unique one for which left invariant vector fields on K are covariantly constant.

Remark 2.12. In Remark 2.3 it was asserted that the process s → χ(t, s) is a Brownian bridge with quadratic co-variation given by χ^{A}(t, ds)χ^{C}(t, ds) = t⟨A, C⟩ds, that is to say Law(χ(t, ·)) is pinned Wiener measure on L(𝔨) with variance t. To check this let p_{t}(x) = (2πt)^{− dim 𝔨/2}e^{−|x|_{𝔨}^{2}/2t} be the Euclidean heat kernel on 𝔨. Then for a cylinder function f on L(𝔨) based on a partition ℙ = {0 = s_{0} < s_{1} < s_{2} < ⋯ < s_{n} < 1}, we must show that

𝔼f(χ(t, ·)) = 𝔼 [f(B)\frac{p_{t(1−α)}(B_{α})}{p_{t}(0)}] , (2.11)

where B_{s} = β(t, s) – a 𝔨 – valued Brownian motion with variance t.

Proof. To prove Eq. (2.11), let Z_{s} = p_{t}(0)^{−1}p_{t(1−s)}(β(t, s)) for 0 ≤ s < 1, then by Itô’s lemma and the fact that

\frac{∂}{∂s}p_{t(1−s)}(x) = −\frac{1}{2}tΔ_{𝔨}p_{t(1−s)}(x) and ∇ log p_{t(1−s)}(x) = −\frac{1}{t(1 − s)}x

we have

dZ_{s} = −Z_{s} \frac{1}{t(1 − s)}⟨B_{s}, dB_{s}⟩ with Z_{0} = 1.

By Girsanov’s theorem (see for example Theorem 20 on p.109 in [27])

M_{s} := B_{s} − ∫_{0}^{s} \frac{1}{Z_{r}} dZ_{r}dB_{r} = B_{s} + ∫_{0}^{s} \frac{1}{(1 − r)}B_{r}dr (2.12)

is a martingale on [0, α] relative to the measure Z_{α}P. Since M has the same quadratic variation as β(t, ·), by Lévy’s criteria, M is a 𝔨 – valued Brownian motion with variance t under the measure Z_{α}P. Interpreting (2.12) as stochastic differential equation for B,

dB = dM − \frac{1}{(1 − s)}B_{s}ds with B_{0} = 0,

we find by variation of parameters that

B_{s} = ∫_{0}^{s} e^{− ∫_{r}^{s} \frac{1}{(1−σ)} dσ}dM_{r} = ∫_{0}^{s} \frac{1 − s}{1 − r}dM_{r}.

This shows that, under Z_{α}P, {B_{s}}_{0≤s≤α} is still a Gaussian process. Moreover, for 0 ≤ σ ≤ s ≤ α,

𝔼[B_{s}^{C}B_{r}^{D}Z_{α}] = 𝔼[(∫_{0}^{s} \frac{1 − s}{1 − r}dM_{r}^{C}) (∫_{0}^{σ} \frac{1 − σ}{1 − r} dM_{r}^{D}) Z_{α}]
= t(1 − s)(1 − σ)⟨C, D⟩ ∫_{0}^{σ} \frac{1}{(1 − r)^{2}} dr
= t(1 − s)(1 − σ) (1 − \frac{1}{1 − σ} ) ⟨C, D⟩
= tσ(1 − s) = tG_{0}(σ, s)⟨C, D⟩

which is the same covariance function as χ(t, ·). Therefore {B_{s}}_{0≤s≤α} under the measure Z_{α}P has the same law as {χ(t, s)}_{0≤s≤α} under the measure P. This is the assertion in Eq. (2.11).

2.3. Heat kernel measure on W (K) and L(K). In this section we are going to define heat kernel measures on W (K) and L(K) by formally replacing K from the previous section by W (K) and L(K) respectively. Following Malliavin [24], we have the following theorem.

Theorem 2.13 (Brownian Motion on W (K) and L(K)). There are jointly continuous solutions Σ(t, s) and Σ^{0}(t, s) to the stochastic differential equations:

Σ(δt, s) = Σ(t, s)β(δt, s) with Σ(0, s) = e ∀s ∈ [0, ∞), (2.13)

and

Σ^{0}(δt, s) = Σ^{0}(t, s)χ(δt, s) with Σ(0, s) = e ∀s ∈ [0, 1]. (2.14)

As before β(δt, s) denotes the Stratonovich differentials of the processes t → β(t, s) and similarly for Σ(δt, s), Σ^{0}(δt, s), and χ(δt, s).

Proof. Such results may be found in Baxendale, [4], Malliavin [24], or in Theorem 3.8 in Driver [13]. The last two references cover the L(K) case, however the proof of the W (K) case is the same, just replace G_{0}(s, σ) by s ∧ σ throughout.

Definition 2.14 (Heat Kernel Measures on W (K) and L(K)). The measures ν_{t} = Law(Σ(t, ·) and ν_{t}^{0} = Law(Σ^{0}(t, ·)) are called heat kernel measures on W (K) and L(K) respectively. So ν_{t} and ν_{t}^{0} are determined by

ν_{t}(f) = 𝔼f(Σ(t, ·)) and ν_{t}^{0}(f) = 𝔼f(Σ^{0}(t, ·)) (2.15)

for all bounded F – measurable f. Notice that ν_{t}^{0}(L(K)) = 1 because Σ^{0}(t, 0) = Σ^{0}(t, 1) = e, P – almost surely.

Corollary 3.10 below justifies calling ν_{t} and ν_{t}^{0} heat kernel measures.

2.4. Statement of Results. The following theorem is Lemma 1 in Airault and Malliavin [1].

Theorem 2.15. Let t > 0, then ν_{t} = μ_{t} on W (K), i.e. heat kernel measure at time t and Wiener measure with variance t are the same on W (K).

This theorem is also proved in Lemma 3.3 of Srimurthy [31]. Since this theorem is crucial to the rest of the paper, we will give a proof in Section 4 below. The following theorem is the main result of the paper.

Theorem 2.16. Let t > 0, then ν_{t}^{0} ≪ μ_{t}^{0}, i.e. heat kernel measure at time t is absolutely continuous relative to pinned Wiener measure with variance t. Moreover, the Radon-Nikodym derivative, dν_{t}^{0}/dμ_{t}^{0}, satisfies the bound

\frac{dν_{t}^{0}}{dμ_{t}^{0}} ≤ e^{C_{t}}

where

C_{t} ≡ log [(2πt)^{\frac{1}{2} dim 𝔨}p_{t}^{K} (e)] . (2.16)

(Standard heat kernel asymptotics shows that lim_{t→0} C_{t} = 0, see Lemma 6.1 below.)

The proof of this theorem (given in Section 6) will be a combination of the maximum principle along with a theorem of Airault and Malliavin [2]. In order to state the Airault-Malliavin theorem, let us recall that the coordinate process π_{s} : L(K) → K (see Definition 2.4) is a semi-martingale relative to pinned Wiener measure, μ_{t}^{0}, see for example Bismut [6] or Theorem 2.3 in [12]. Hence we may define the 𝔨 – valued semi-martingale{b_{s}}_{0≤s≤1} by

b_{s} := ∫_{0}^{s} π_{r}^{−1}δπ_{r}. (2.17)

Remarks 2.17. i) Technically speaking the stochastic integral in Eq. (2.17) depends on the measure μ_{t}^{0} and in particular on t > 0. So a more appropriate notation would be to display this t dependence and write b_{s}^{t} for the μ_{t}^{0} – a.e. defined stochastic integral ∫_{0}^{s} π_{r}^{−1}δπ_{r}. Since we will only need the process b_{s} for one fixed value of t, we will stick with the notation in Eq. (2.17).

ii) Gross shows (see Lemma 4.8 and Remark 4.9 in [20]) that b_{1} ∈ L^{p}(L(K), μ_{t}^{0}) and that b_{s} → b_{1} in L^{p}(L(K), μ_{t}^{0}) as s → 1 for all 1 ≤ p < ∞.

Theorem 2.18 (Airault & Malliavin). Let V_{t} : L(K) → ℝ be the “potential,”

V_{t} = \frac{1}{2t^{2}} |b_{1}|_{𝔨}^{2} − c_{t} (2.18)

where b_{1} is defined in Eq. (2.17) and

c_{t} ≡ \frac{dC_{t}}{dt} = \frac{dim 𝔨}{2t} + ∂_{t} log p_{t}^{K} (e) . (2.19)

Then for any smooth cylindrical function f : L (K) → ℝ (see Definition 2.4)

∂_{t}μ_{t}^{0}(f) = μ_{t}^{0} [(\frac{1}{2}△_{L(K)} + V_{t}) f] , (2.20)

where △_{L(K)} is the generator of the process Σ^{0}(t, ·), see Definition 3.6 and Proposition 3.9 below.

We will give a simplified (in our view) proof of this theorem in Section 5. The proof relies on Theorem 2.15 and integration by parts on (W (K), μ_{t}).

3. Generators of Σ(t, ·) and Σ^{0}(t, ·)

Much of the material in this section may be found in [18] and [13]. Nevertheless, in order to introduce the notation and for the readers convenience we will summarize some of the results in these papers.

3.1. Cameron-Martin spaces.

Definition 3.1. Given a continuous function h : [0, 1] → 𝔨 define

(h, h)_{H} = \{ \begin{matrix}∫_{0}^{1} |h′ (s)|^{2} ds & if & h is absolutely continuous \\ ∞ & otherwise. \end{matrix}

The Cameron–Martin space of 𝔨 is

H ≡ {h ∈ C([0, 1] → 𝔨)|h (0) = 0 and (h, h) < ∞}

which we equip with the inner product

(h, k) = ∫_{0}^{1}⟨h′ (s) , k′(s)⟩ds.

The pinned Cameron–Martin space is

H_{0} ≡ {h ∈ H (𝔨) |h (1) = 0}

which is a closed subspace of H. (The Hilbert spaces H and H_{0} are to be thought of as the “Lie algebras” to the groups W (K) and L (K).)

Notation 3.2. Let S ⊂ H and S_{0} ⊂ H_{0} be orthonormal bases for H and H_{0} respectively.

Lemma 3.3. Let 𝔨_{0} ⊂ 𝔨 be an orthonormal basis for 𝔨, G(s, t) = s∧t and G_{0}(s, t) ≡ s ∧ t − st for all s, t ∈ [0, 1]. Then

(3.1)  ∑_{h∈S} h(s) ⊗ h(t) = G(s, t) ∑_{A∈𝔨_{0}} A ⊗ A ∈ 𝔨 ⊗ 𝔨.
(3.2)  ∑_{h∈S_{0}} h(s) ⊗ h(t) = G_{0}(s, t) ∑_{A∈𝔨_{0}} A ⊗ A ∈ 𝔨 ⊗ 𝔨.

Proof. Let A, B ∈ 𝔨. Since G(t, ·)B and G(s, ·)A are in H,

(G(t, ·)B, G(s, ·)A) = ∑_{h∈S}(G(t, ·)B, h)(h, G(s, ·)A) (3.3)

where the sum is absolutely convergent. By the fundamental theorem of calculus, G satisfies the reproducing property,

∫_{0}^{1} ∂_{s}G(t, s)h′(s)ds = h(t) for all h ∈ H.

Combined this equation with Eq. (3.3) shows that

G(s, t)⟨B, A⟩ = ∑_{h∈S}⟨B, h(t)⟩⟨h(s), A⟩

which implies Eq. (3.1) since A and B are arbitrary. Equation (3.2) is proved similarly, see Lemma 3.8 in [18] for more details.

3.2. Derivatives and Laplacians on L(K) and W (K).

Definition 3.4 (Left invariant derivatives). Given h ∈ H (or H_{0}) and f : W (K) → ℝ (or f : L(K) → ℝ) a smooth cylinder function, define

(∂_{h}f) (σ) := \frac{d}{dt}|_{0}f(σe^{th}) for all σ ∈ W (K) (σ ∈ L(K))

where σe^{th} ∈ W (K) (σe^{th} ∈ L(K)) is defined by (σe^{th}) (s) := σ(s)e^{th(s)} for s ∈ [0, 1].

Remark 3.5. Suppose that f = F ∘π_{ℙ} where ℙ = {0 = s_{0} < s_{1} < s_{2} < ⋯ < s_{n} < 1} is a partition of [0, 1] and F : K^{n} → ℝ is a smooth function. For A ∈ 𝔨 and i ∈ {1, 2, …, n}, let

A^{(i)}F (x_{1}, x_{2}, …, x_{n}) = \frac{d}{dt}|_{0}F (x_{1}, x_{2}, …, x_{i−1}, x_{i}e^{tA}, x_{i+1}, …, x_{n}),

so that A^{(i)} is the action of A on the i^{th} variable of F. Then for h ∈ H (or h ∈ H_{0}),

∂_{h}f = ∑_{i=1}^{n} (h(s_{i})^{(i)}F ) ∘ π_{ℙ}. (3.4)

In particular ∂_{h}f is still a smooth cylinder function. Therefore the operator ∂_{h}^{2}f is well defined and is given by

∂_{h}^{2}f = ∑_{i,j=1}^{n} (h(s_{j})^{(j)}h(s_{i})^{(i)}F ) ∘ π_{ℙ}. (3.5)

Definition 3.6. Again suppose that f = F ∘ π_{ℙ} is a smooth cylinder function as in Definition 2.4. Define the Laplacians on W (K) and L (K)) by

△_{W (K)}f ≡ ∑_{h∈S} ∂_{h}^{2}f and
△_{L(K)}f ≡ ∑_{h∈S_{0}} ∂_{h}^{2}f

respectively.

Remark 3.7. Combining Eqs. (3.1), (3.2) and (3.5) we find

△_{W (K)}f = ∑_{h∈S} ∑_{i,j=1}^{n} (h(s_{j})^{(j)}h(s_{i})^{(i)}F ) ∘ π_{ℙ}

(3.6)  = ∑_{A∈𝔨_{0}} ∑_{i,j=1}^{n} G(s_{i}, s_{j}) (A^{(j)}A^{(i)}F ) ∘ π_{ℙ}

and

△_{L(K)}f = ∑_{h∈S_{0}} ∑_{i,j=1}^{n} (h(s_{j})^{(j)}h(s_{i})^{(i)}F ) ∘ π_{ℙ}

(3.7)  = ∑_{A∈𝔨_{0}} ∑_{i,j=1}^{n} G_{0}(s_{i}, s_{j}) (A^{(j)}A^{(i)}F ) ∘ π_{ℙ}.

Notation 3.8. Given, ℙ = {0 = s_{0} < s_{1} < s_{2} < ⋯ < s_{n} < 1}, a partition of [0, 1] and F ∈ C^{∞}(K^{n}), let

L_{ℙ}F = ∑_{A∈𝔨_{0}} ∑_{i,j=1}^{n} G(s_{i}, s_{j})A^{(j)}A^{(i)}F (3.8)

and

L_{ℙ}^{0}F = ∑_{A∈𝔨_{0}} ∑_{i,j=1}^{n} G_{0}(s_{i}, s_{j})A^{(j)}A^{(i)}F. (3.9)

With this notation we may write Eqs. (3.6) and (3.7) as

△_{W (K)}(F ∘ π_{ℙ}) = (L_{ℙ}F) ∘ π_{ℙ} and △_{L(K)}(F ∘ π_{ℙ}) = (L_{ℙ}^{0}F ) ∘ π_{ℙ}. (3.10)

3.3. Heat equations.

Proposition 3.9. The processes Σ(t, ·) and Σ^{0}(t, ·) are diffusion processes with △_{W (K)} and △_{L(K)} as generators. More precisely, if f = F ∘ π_{ℙ} is a cylinder function as above, then

M_{t}^{f} = f(Σ(t, ·)) − f(e) − \frac{1}{2} ∫_{0}^{t} (Δ_{W (K)}f) (Σ(τ, ·))dτ (3.11)

and

M_{t}^{f} = f(Σ^{0}(t, ·)) − f(e) − \frac{1}{2} ∫_{0}^{t} (Δ_{L(K)}f) (Σ^{0}(τ, ·))dτ (3.12)

are martingales.

Proof. We will only prove Eq. (3.11) since the proof of Eq. (3.12) is completely analogous. Let Σ_{ℙ}(t) := π_{ℙ}(Σ(t, ·)) ∈ K^{n} and B_{ℙ}(t) = (β(t, s_{1}), …, β(t, s_{n})) , then f(Σ(t, ·)) = F (Σ_{ℙ}(t)) and by Itô’s Lemma we have that

df(Σ(t, ·)) = dF (Σ_{ℙ}(t))
= ∑_{i=1}^{n} ∑_{A∈𝔨_{0}} A^{(i)}F (Σ_{ℙ}(t))β^{A}(δt, s_{i})

= ∑_{i=1}^{n} ∑_{A∈𝔨_{0}} A^{(i)}F (Σ_{ℙ}(t))β^{A}(dt, s_{i})

+\frac{1}{2} ∑_{i,j=1}^{n} ∑_{A,B∈𝔨_{0}} B^{(j)}A^{(i)}F (Σ_{ℙ}(t))β^{A}(dt, s_{i})β^{B}(dt, s_{j})

= ∑_{i=1}^{n} ∑_{A∈𝔨_{0}} A^{(i)}F (Σ_{ℙ}(t))β^{A}(dt, s_{i})

+\frac{1}{2} ∑_{i,j=1}^{n} ∑_{A∈𝔨_{0}} G(s_{i}, s_{j})A^{(j)}A^{(i)}F (Σ_{ℙ}(t))dt

= ∑_{i=1}^{n} ∑_{A∈𝔨_{0}} A^{(i)}F (Σ_{ℙ}(t))β^{A}(dt, s_{i}) + \frac{1}{2} (Δ_{W (K)}f) (Σ(t, ·))dt.

This shows that M_{t}^{f} is the martingale;

M_{t}^{f} = ∑_{i=1}^{n} ∑_{A∈𝔨_{0}} ∫_{0}^{t} A^{(i)}F (Σ_{ℙ}(τ))β^{A}(dτ, s_{i}).

Corollary 3.10. The measures ν_{t} and ν_{t}^{0} satisfy the heat equations on W (K) and L(K) in the following weak sense. If f : W (K) → ℝ is a smooth cylinder function then

∂_{t}ν_{t}(f) = \frac{1}{2}ν_{t}(Δ_{W (K)}f) (3.13)

and

∂_{t}ν_{t}^{0}(f) = \frac{1}{2}ν_{t}^{0}(Δ_{L(K)}f). (3.14)

Proof. Taking expectations of Eq. (3.11) shows that

0 = 𝔼M_{t}^{f} = 𝔼f(Σ(t, ·)) − f(e) − \frac{1}{2} ∫_{0}^{t} 𝔼 (Δ_{W (K)}f) (Σ(τ, ·))dτ
= ν_{t}(f) − f(e) − \frac{1}{2} ∫_{0}^{t} ν_{τ} (Δ_{W (K)}f) dτ.

Differentiating this equation in t proves Eq. (3.13). Eq. (3.14) is proved analogously.

Corollary 3.11 (Heat solution). Suppose that u : L(K) → ℝ is a smooth cylinder function and let

H(t, σ) = ∫_{L(K)} u(σγ^{−1})dν_{t}^{0}(γ), (3.15) then

∂_{t}H(t, σ) = \frac{1}{2}△_{L(K)}H(t, σ) and lim_{t→0} H(t, σ) = u(σ) (3.16)

Proof. For σ ∈ L(K), let u_{σ} : L(K) → ℝ be the cylinder function defined by u_{σ}(γ) = u(σγ^{−1}). Notice that for h ∈ H,

(∂_{h}u_{σ}) (γ) = \frac{d}{dϵ}|_{0}u_{σ}(γe^{ϵh}) = \frac{d}{dϵ}|_{0}u(σe^{−ϵh}γ^{−1}) = −∂_{h}(σ → u_{σ}(γ))

and therefore

(Δ_{L(K)}u_{σ}) (γ) = Δ_{L(K)}(σ → u_{σ}(γ)).

Thus by Corollary 3.10,

∂_{t}H(t, σ) = \frac{1}{2} ∫_{L(K)} (△_{L(K)}u_{σ}) (γ)dν_{t}^{0}(γ)
= \frac{1}{2} ∫_{L(K)} Δ_{L(K)}(σ → u_{σ}(γ))dν_{t}^{0}(γ)

= \frac{1}{2}Δ_{L(K)} (σ → ∫_{L(K)} u_{σ}(γ)dν_{t}^{0}(γ))
= \frac{1}{2}△_{L(K)}H(t, σ).

Working with the explicitly representation of u as a cylinder function and using Eq. (3.10), it is easy to justify the interchange of Δ_{L(K)} with the integral in the third equality. This proves the first assertion in Eq. (3.16). The second follows from the dominated convergence theorem and the identity,

H(t, σ) = 𝔼 [u(σΣ^{0}(t, ·)^{−1})] ,

where Σ^{0}(t, s) is the process defined in Eq. (2.14) of Theorem 2.13.

4. The path group case

In the next subsection we will give a proof of Theorem 2.15. However, before doing this let us record the following trivial Corollary of Theorem 2.15 and Corollary 3.10 above. This corollary will be key to our proof of the Airault Malliavin theorem in Section 5.

Corollary 4.1. The Wiener measure μ_{t} with variance t satisfies (weakly) the heat equation on W (K), i.e. if f : W (K) → ℝ is a smooth cylinder function then

∂_{t}μ_{t}(f) = \frac{1}{2}μ_{t}(Δ_{W (K)}f). (4.1)

4.1. Proof of Theorem 2.15.

Proof. As mentioned in Section 2, the reader may find this theorem in Lemma 1 of Airault and Malliavin [1] or Lemma 3.3 of Srimurthy [31]. It would also be possible to give a proof using two parameter stochastic calculus as developed in Norris [26]. Rather than introduce this machinery, we will give a more pedestrian but perhaps less illuminating proof. Our proof is similar to that in [31].

Let Σ denote the process defined in Theorem 2.13 and ℙ = {0 = s_{0} < s_{1} < s_{2} < ⋯ < s_{n} < 1} be a partition of [0, 1]. Let

U_{i}(t) := Σ(t, s_{i})Σ(t, s_{i−1})^{−1} (4.2)

and

B_{i}(t) := ∫_{0}^{t} Ad_{Σ(τ,s_{i−1})} (β(δτ, s_{i}) − β(δτ, s_{i−1})) (4.3)

for i = 1, 2, …, n. By Eq. (2.13) and Itô’s Lemma,

δ_{t}Σ(t, s)^{−1} = −β(δt, s)Σ(t, s)^{−1}

and therefore

δU_{i}(t) = Σ(t, s_{i}) (β(δt, s_{i}) − β(δt, s_{i−1})) Σ(t, s_{i−1})^{−1}
= U_{i}(t)Ad_{Σ(t,s_{i−1})} (β(δt, s_{i}) − β(δt, s_{i−1}))
(4.4)  = U_{i}(t)δB_{i}(t).

Because, t → β(t, s_{i−1}) and t → β(t, s_{i}) − β(t, s_{i−1}) are independent Brownian motions on 𝔨,

Ad_{Σ(t,s_{i−1})} (β(δt, s_{i}) − β(δt, s_{i−1}))

= Ad_{Σ(t,s_{i−1})} (β(dt, s_{i}) − β(dt, s_{i−1}))

+ \frac{1}{2}Ad_{Σ(t,s_{i−1})}[β(dt, s_{i−1}), (β(dt, s_{i}) − β(dt, s_{i−1}))]_{𝔨}

= Ad_{Σ(t,s_{i−1})} (β(dt, s_{i}) − β(dt, s_{i−1})) .

Therefore the Stratonovich differentials in Eq. (4.3) may be replaced by Itô differentials to learn that B_{i}(t) is the martingale

B_{i}(t) := ∫_{0}^{t} Ad_{Σ(τ,s_{i−1})} (β(dτ, s_{i}) − β(dτ, s_{i−1})) .

Claim. The processes B_{1}, B_{2}, … B_{n} are independent 𝔨 – valued Brownian motions with variances Δ_{i}s := s_{i} − s_{i−1} for i = 1, 2, …, n.

To prove this claim, let C, D ∈ 𝔨, and let B_{i}^{C}(t) = ⟨B_{i}(t), C⟩, B_{j}^{D}(t) = ⟨B_{j}(t), D⟩ and Δ_{i}β(t) := β(t, s_{i}) − β(t, s_{i−1}). Then because ⟨·, ·⟩ is Ad_{K} – invariant,

dB_{i}^{C}(t) = ⟨Ad_{Σ(t,s_{i−1})}dΔ_{i}β(t), C⟩ = ⟨dΔ_{i}β(t), Ad_{Σ(t,s_{i−1})}^{−1}C⟩.

Thus the differential of the quadratic co-variation of B_{i}^{C} and B_{j}^{D} is given by,

dB_{i}^{C}(t)dB_{j}^{D}(t) = ⟨dΔ_{i}β(t), Ad_{Σ(t,s_{i−1})}^{−1}C⟩⟨dΔ_{j}β(t), Ad_{Σ(t,s_{j−1})}^{−1}D⟩
= ∑_{A∈𝔨_{0}}⟨A, Ad_{Σ(t,s_{i−1})}^{−1}C⟩⟨A, Ad_{Σ(t,s_{j−1})}^{−1}D⟩ dΔ_{i}β^{A}(t) dΔ_{j}β^{A}(t)
= δ_{ij} ∑_{A∈𝔨_{0}}⟨A, Ad_{Σ(t,s_{i−1})}^{−1}C⟩⟨A, Ad_{Σ(t,s_{i−1})}^{−1}D⟩Δ_{i}sdt
= δ_{ij}⟨Ad_{Σ(t,s_{i−1})}^{−1}C, Ad_{Σ(t,s_{i−1})}^{−1}D⟩Δ_{i}sdt
(4.5)  = δ_{ij}⟨C, D⟩Δ_{i}sdt,

wherein the third equality we have used: i) Δ_{i}β^{A}(·) = β^{A}(·, s_{i}) − β^{A}(·, s_{i−1}) and Δ_{j}β^{A}(·) = β^{A}(·, s_{j}) − β^{A}(·, s_{j−1}) are independent if i ≠ j and Δ_{i}β^{A}(·) is a 𝔨 – valued Brownian motion with variance Δ_{i}s. In the last equality we again have used the Ad_{K} – invariance of ⟨·, ·⟩. Eq. (4.5) along with Lévy’s criteria proves the claim.

Since the U_{i}’s in Eq. (4.2) satisfy Eq. (4.4), the claim implies that U_{1}(t), U_{2}(t), …, U_{n}(t) are independent K – valued Brownian motion with variance Δ_{1}s, Δ_{2}s, …, Δ_{n}s respectively. Suppose that f = F ∘ π_{ℙ} is a bounded cylinder function on W (K). Define F̃ : K^{n} → ℝ so that

F (x_{1}, x_{2}, x_{3}, …, x_{n}) = F̃(x_{1}, x_{2}x_{1}^{−1}, x_{3}x_{2}^{−1} …, x_{n}x_{n−1}^{−1})

for all x_{i} ∈ K. Then

f(Σ(t, ·)) = F̃(U_{1}(t), U_{2}(t), …, U_{n}(t))

and therefore

ν_{t}(f) = 𝔼f(Σ(t, ·)) = 𝔼F̃(U_{1}(t), U_{2}(t), …, U_{n}(t))
(4.6)  = ∫_{K^{n}} F̃(x_{1}, ⋯, x_{n}) ∏_{i=n}^{n} p_{tΔ_{i}s}^{K}(x_{i})dx_{i}.

Let x_{0} := e. Using the invariance of Haar measure, make the translations

x_{2} → x_{2}x_{1}^{−1} then
x_{3} → x_{3}x_{2}^{−1} then
⋮
x_{n} → x_{n}x_{n−1}^{−1}

in the last integral of Eq. (4.6) to find

ν_{t}(f) = ∫_{K^{n}} F̃(x_{1}, x_{2}x_{1}^{−1} ⋯, x_{n}x_{n−1}^{−1}) ∏_{i=n}^{n} p_{Δ_{i}st}^{K}(x_{i}x_{i−1}^{−1})dx_{i}

= ∫_{K^{n}} F (x_{1}, x_{2}, x_{3}, …, x_{n}) ∏_{i=n}^{n} p_{Δ_{i}st}^{K}(x_{i}x_{i−1}^{−1})dx_{i}

(4.7)  = ∫_{K^{n}} F (x_{1}, x_{2}, x_{3}, …, x_{n}) ∏_{i=n}^{n} p_{Δ_{i}st}^{K}(x_{i−1}^{−1}x_{i})dx_{i}

wherein the last equality we have use the fact that p_{t}^{K}(·) is a class function, see Remark 2.10.

Comparing Eq. (4.7) with Eq. (2.8), shows that ν_{t}(f) = μ_{t}(f) for all bounded cylinder functions f on W (K) which implies that ν_{t} = μ_{t} by Lemma 2.7.

5. Proof of the Airault-Malliavin Theorem 2.18

This subsection is devoted to the proof of Theorem 2.18. We will need some, mostly well known, preliminary results regarding integration by parts on (W (K), μ_{t}). These results will be gathered in the next subsection.

5.1. Integration by parts and strong differentiability. The key result here for the remainder of the paper is Corollary 5.6. The reader may skip this subsection if she/he is willing to accept Corollary 5.6 below.

Definition 5.1. Let L^{∞−}(W (K), μ_{t}) = ∩_{1≤p<∞}L^{p}(W (K), μ_{t}) and h ∈ H. A function f ∈ L^{∞−}(W (K), μ_{t}) is said to be strongly h differentiable provided there is a function g ∈ L^{∞−}(W (K), μ_{t}) such that

g = L^{p}(μ_{t})– lim_{ϵ→0} \frac{f(σe^{ϵh}) − f(σ)}{ϵ}

for all 1 ≤ p < ∞. We will denote the function g, if it exists, by ∂_{h}f.

Cylinder functions are strongly h – differentiable for all h ∈ H and ∂_{h}f is given by Eq. (3.4). Another example is given in Lemma 5.5 below.

Definition 5.2. An element k ∈ W (K) is a finite energy path if

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

Letting k ∈ W (K) be a finite energy path and b_{s} being as in Eq. (2.17), then for μ_{t} – a.e. σ ∈ W (K),

b_{s}(σk) = ∫_{0}^{s} (σ(r)k(r))^{−1} δ (σk) (r)
= ∫_{0}^{s} k^{−1}(r)σ^{−1}(r)[δσ(r)k(r) + σ(r)k′(r) dr]
(5.1)  = ∫_{0}^{s} Ad_{k^{−1}(r)}db_{r}(σ) + ∫_{0}^{s} k^{−1}(r)k′(r) dr.

Since Ad_{k^{−1}(r)} is orthogonal on 𝔨, Lévy’s characterization of Brownian motion shows that B_{s} := ∫_{0}^{s} Ad_{k^{−1}(r)}db_{r} on (W (K), μ_{t}) is still a Brownian motion with variance t. This observation and the Cameron-Martin theorem is essentially the proof of the following quasi invariance theorem of Albeverio and Hoegh-Krohn, see [3], [29], and [28].

Theorem 5.3 (Albeverio & Hoegh-Krohn). Let k be a finite-energy path on K and f : W (K) → ℝ be a bounded measurable function. Then

∫_{W (K)} f(σ)dμ_{t}(σ) = ∫_{W (K)} f(σk)J_{k}(σ)dμ_{t}(σ), (5.2)

where

J_{k} := exp (−\frac{1}{t} ∫_{0}^{1}⟨k′(s)k^{−1}(s), db_{s}⟩ − \frac{1}{2t} ∫_{0}^{1} |k′(s)k^{−1}(s)|^{2} ds) . (5.3)

Proof. Let h(s) := ∫_{0}^{s} k^{−1}(r)k′(r) dr, B_{s} := ∫_{0}^{s} Ad_{k^{−1}(r)}db_{r}, and f̃ be a measurable function on C([0, 1] → 𝔨) such that f̃(b_{·}(σ)) = f(σ) for μ_{t} – a.e. σ. Using the Ad_{K} – invariance of the inner product ⟨·, ·⟩ on 𝔨, we have

∫_{0}^{1} |k′(r)k^{−1}(r)|^{2} dr = ∫_{0}^{1} |k^{−1}(r)k′(r)|^{2} dr

and

∫_{0}^{1}⟨k′(r)k^{−1}(r), db_{r}⟩ = ∫_{0}^{1}⟨k′(r)k^{−1}(r), Ad_{k(r)}dB_{r}⟩
= ∫_{0}^{1}⟨Ad_{k^{−1}(r)}k′(r)k^{−1}(r), dB_{r}⟩
= ∫_{0}^{1}⟨k^{−1}(r)k′(r), dB_{r}⟩.

Combining these equations show that J_{k} may be written as

J_{k} := exp (−\frac{1}{t} ∫_{0}^{1}⟨h′, dB_{r}⟩ − \frac{1}{2t} ∫_{0}^{1} |h′(r)|^{2} dr) . (5.4)

By Eq. (5.1),

∫_{W (K)} f(σk)J_{k}(σ)dμ_{t}(σ) = ∫_{W (K)} f̃(b_{·}(σk))J_{k}(σ)dμ_{t}(σ)
= ∫_{W (K)} f̃(B_{·}(σ) + h)J_{k}(σ)dμ_{t}(σ)
(5.5)  = ∫_{W (K)} f̃(B_{·}(σ))dμ_{t}(σ)

wherein the last equality we have used the Cameron-Martin (or Girsanov’s) theorem. Since B and b have the same laws, being 𝔨 – valued Brownian motions with variance t,

∫_{W (K)} f̃(B_{·}(σ))dμ_{t}(σ) = ∫_{W (K)} f̃(b_{·}(σ))dμ_{t}(σ) = ∫_{W (K)} f(σ)dμ_{t}(σ). (5.6)

Combining Eqs. (5.4), (5.5) and (5.6) proves the theorem.

Corollary 5.4. Let h ∈ H (𝔨) and suppose that f and g are strongly h – differentiable, then

μ_{t}(g∂_{h}f) = μ_{t}((−∂_{h}g + j_{h}g)f) (5.7)

where

j_{h} := \frac{1}{t} ∫_{0}^{1} ⟨h′ (s) , db_{s}⟩ . (5.8)

This corollary has been proved in the more general context of Wiener measure on a Riemannian manifold in Driver [11].

Proof. Let k = e^{ϵh} and replace f by fg in Eq. (5.2) of Theorem 5.3 to find

μ_{t}(fg) = ∫_{W (K)} f(σe^{ϵh})g(σe^{ϵh})J_{e^{ϵh}}(σ)dμ_{t}(σ).

Differentiate this equation in ϵ implies

0 = μ_{t} (∂_{h}f · g + f∂_{h}g + fg \frac{d}{dϵ}|_{0}J_{e^{ϵh}})

which proves the corollary provided that

\frac{d}{dϵ}|_{0}J_{e^{ϵh}} = −j_{h} in L^{p}(μ_{t}) for all p ∈ [1, ∞).

We will not carry out the convergence details here which are fairly routine. The interested reader may refer to Gross [21] or Section 9 in [11]. However, let us check “algebraically” that the formula in Eq. (5.8) is correct. Computing \frac{d}{dϵ}|_{0}J_{e^{ϵh}} gives

\frac{d}{dϵ}|_{0}J_{e^{ϵh}} = \frac{d}{dϵ}|_{0} exp (−\frac{1}{t} ∫_{0}^{1}⟨(\frac{d}{dr} e^{ϵh(r)})e^{−ϵh(r)}, db_{r}⟩ − \frac{1}{2t} ∫_{0}^{1} |(\frac{d}{dr} e^{ϵh(r)})e^{ϵh(r)}|^{2} dr)

= −\frac{1}{t} \frac{d}{dϵ}|_{0} [∫_{0}^{1}⟨(\frac{d}{dr} e^{ϵh(r)})e^{−ϵh(r)}, db_{r}⟩ − \frac{1}{2t} ∫_{0}^{1} |(\frac{d}{dr} e^{ϵh(r)})e^{ϵh(r)}|^{2} dr]

= −\frac{1}{t} ∫_{0}^{1}⟨h′(r), db_{r}⟩.

because

\frac{d}{dϵ}|_{0}(\frac{d}{dr} e^{ϵh(r)})e^{−ϵh(r)} = h′(r)

and

\frac{d}{dϵ}|_{0} |(\frac{d}{dr} e^{ϵh(r)})e^{ϵh(r)}|^{2} = 2 ⟨(\frac{d}{dr} e^{ϵh(r)})e^{ϵh(r)}|_{ϵ=0}, \frac{d}{dϵ}|_{0}(\frac{d}{dr} e^{ϵh(r)})e^{ϵh(r)}⟩
= 2 ⟨0, \frac{d}{dϵ}|_{0}(\frac{d}{dr} e^{ϵh(r)})e^{ϵh(r)}⟩ = 0.

Lemma 5.5. For each h ∈ H, the function j_{h} is strongly h differentiable and

∂_{h}j_{h} = \frac{1}{t} ∫_{0}^{1} ⟨ad_{h(r)}h′(r), db_{r}(σ)⟩ + \frac{1}{t} ∫_{0}^{1} |h′(r)|^{2}dr. (5.9)

Proof. According to Eq. (5.1),

j_{h}(σe^{ϵh}) = \frac{1}{t} ∫_{0}^{1} ⟨h′(r), Ad_{e^{−ϵh(r)}}db_{r}(σ) + e^{−ϵh(r)} \frac{d}{dr} e^{ϵh(r)} dr⟩
= \frac{1}{t} ∫_{0}^{1} ⟨Ad_{e^{ϵh(r)}}h′(r), db_{r}(σ)⟩ + \frac{1}{t} ∫_{0}^{1} ⟨h′(r), e^{−ϵh(r)} \frac{d}{dr} e^{ϵh(r)} dr⟩ .

Therefore, again ignoring convergence questions,

∂_{h}j_{h}(σ) = \frac{d}{dϵ}|_{0}j_{h}(σe^{ϵh}) = \frac{1}{t} ∫_{0}^{1} ⟨ad_{h(r)}h′(r), db_{r}(σ)⟩ + \frac{1}{t} ∫_{0}^{1} |h′(r)|^{2}dr.

Here the convergence questions are even easier since we only have jointly Gaussian random variables to contend with and L^{2} – convergence of Gaussian random variables implies L^{p} convergence for p < ∞. The reader may find more details in Section 4 in Gross [20].

The following Corollary is a key ingredient in our proof of the Airault – Malliavin Theorem 2.18.

Corollary 5.6. Let f be a smooth cylinder function (see Definition 2.4) and h ∈ H (𝔨) such that the Lie bracket [h (s) , h′ (s)] = 0 for a.e. s. Then

μ_{t}(∂_{h}^{2}f) = μ_{t} ([j_{h}^{2} − \frac{1}{t} ∫_{0}^{1} |h′(r)|^{2}dr] f) ,

where j_{h}, is as in Eq. (5.8).

Proof. Two applications of Corollary 5.4 gives

μ_{t}(∂_{h}^{2}f) = μ_{t}(j_{h}∂_{h}f) = μ_{t}((−∂_{h}j_{h} + j_{h}^{2}) f)

which combined with Eq. (5.9) of Lemma 5.5 proves the Corollary.

5.2. Proof of Theorem 2.18.

Proof. Let f = F ∘ π_{ℙ} be a cylinder function on L(K) (see Definition 2.4) and let α ∈ (s_{n}, 1). (We will eventually let α → 1.) Recall the definition of pinned Wiener measure μ_{t}^{0} (see Definition 2.11) says that μ_{t}^{0}(f) = μ_{t}(fη_{t}) where η_{t} := p_{t(1−α)}^{K}(π_{α})/p_{t}^{K}(e). Therefore, by Corollary 4.1,

∂_{t}μ_{t}^{0} (f) = ∂_{t}μ_{t} (fη_{t})
= μ_{t} (f∂_{t}η_{t}) + \frac{1}{2}μ_{t} (△_{W (K)} (fη_{t}))
(5.10)  = I_{α} + J_{α}.

Now

I_{α} = μ_{t}(f∂_{t}η_{t})
(5.11)  = \frac{1}{2p_{t}^{K}(e)}(1 − α)μ_{t} (fΔ_{K}p_{t(1−α)}^{K}(π_{α})) − ∂_{t} ln p_{t}^{K}(e)μ_{t}^{0}(f).

By Eq. (2.8),

μ_{t}(fΔ_{K}p_{t(1−α)}^{K}(π_{α})) = ∫_{K} G(α, x)Δ_{K}p_{t(1−α)}^{K}(x)dx
= ∫_{K} Δ_{K}G(α, x)p_{t(1−α)}^{K}(x)dx

where

G(α, x) := ∫_{K^{n}} F (x_{1}, ⋯, x_{n}) p_{t(α−s_{n})}^{K}(x_{n}^{−1}x) ∏_{i=1}^{n} p_{t△_{i}s}^{K} (x_{i−1}^{−1}x_{i}) dx_{i}.

From this expression we see that Δ_{K}G(α, x) remains bounded as α → 1, so that letting α → 1 in Eq. (5.11) gives

lim_{α→1} I_{α} = −μ_{t}^{0}(f∂_{t} log p_{t}^{K} (e)). (5.12)

We proceed to work on the second term, J_{α}, in Eq. (5.10). Let ℙ_{α} be the partition of [0, 1],

ℙ_{α} = {0 = s_{0} < s_{1} < s_{2} < ⋯ < s_{n} < α < 1},

and set s_{n+1} = α. Define G_{0}^{α}(s, t) = (s ∧ t − α^{−1}st) so that

G(s, t) = s ∧ t = G_{0}^{α}(s, t) + α^{−1}st.

Let η̃_{t}(x_{1}, x_{2}, …, x_{n+1}) = p_{t(1−α)}^{K}(x_{n+1})/p_{t}^{K}(e) and by abuse of notation use F again to denote the function (x_{1}, x_{2}, …, x_{n+1}) ∈ K^{n+1} → F (x_{1}, x_{2}, …, x_{n}). Then by Eqs. (3.10) and (3.8) applied to the partition ℙ_{α},

Δ_{W (K)}(fη_{t}) = L_{ℙ_{α}}(F η̃_{t}) ∘ π_{ℙ_{α}}
= ∑_{i,j=1}^{n+1} ∑_{A∈𝔨_{0}} G_{0}^{α} (s_{i}, s_{j}) A^{(i)}A^{(j)}(F η̃_{t}) ∘ π_{ℙ_{α}}

+ ∑_{i,j=1}^{n+1} ∑_{A∈𝔨_{0}} α^{−1}s_{i}s_{j}A^{(i)}A^{(j)}(F η̃_{t}) ∘ π_{ℙ_{α}}

(5.13)  = S_{α} + T_{α}.

Now for A ∈ 𝔨, let

h_{A}^{α}(s) := α^{−1/2}(s ∧ α)A.

Then by Eq. (3.5),

T_{α} = ∑_{A∈𝔨_{0}} ∂_{h_{A}^{α}}^{2}(fη_{t}). (5.14)

For the S_{α} term in Eq. (5.13), notice that by construction G_{0}^{α}(s, t) = 0 if s or t is in {0, α}. Therefore G_{0}^{α} (s_{i}, s_{j}) = 0 if i or j = n + 1 (i.e. s_{i} or s_{j} is α) so that

S_{α} = ∑_{i,j=1}^{n} ∑_{A∈𝔨_{0}} G_{0}^{α} (s_{i}, s_{j}) A^{(i)}A^{(j)}(F η̃_{t}) ∘ π_{ℙ_{α}}

(5.15)  = η̃_{t} ∘ π_{ℙ_{α}} · ∑_{i,j=1}^{n} ∑_{A∈𝔨_{0}} G_{0}^{α} (s_{i}, s_{j}) (A^{(i)}A^{(j)}F) ∘ π_{ℙ}.

Taking the μ_{t} expectation of Eq. (5.13) and making use of Eq. (5.14) and (5.15) shows that J_{α} from Eq. (5.10) satisfies

J_{α} = \frac{1}{2}μ_{t}^{0} ( ∑_{i,j=1}^{n} ∑_{A∈𝔨_{0}} G_{0}^{α} (s_{i}, s_{j}) (A^{(i)}A^{(j)}F) ∘ π_{ℙ})

+\frac{1}{2}μ_{t} ( ∑_{A∈𝔨_{0}} ∂_{h_{A}^{α}}^{2}(fη_{t}))

= J_{α}^{(1)} + J_{α}^{(2)}.

Since G_{0}^{α} → G_{0} as α → 1,

lim_{α→1} J_{α}^{(1)} = \frac{1}{2}μ_{t}^{0} ( ∑_{i,j=1}^{n} ∑_{A∈𝔨_{0}} G_{0} (s_{i}, s_{j}) (A^{(i)}A^{(j)}F) ∘ π_{ℙ})

= \frac{1}{2}μ_{t}^{0} ( ∑_{i,j=1}^{n+1} ∑_{A∈𝔨_{0}} G_{0} (s_{i}, s_{j}) (A^{(i)}A^{(j)}F) ∘ π_{ℙ})

(5.16)  = \frac{1}{2}μ_{t}^{0} (Δ_{L(K)}f) ,

where we have used Eq. (3.10) for the last equality.

HEAT KERNEL VS. PINNED WIENER MEASURE 21

By Corollary 5.6,

2J_{α}^{(2)} = ∑_{A∈𝔨_{0}} μ_{t} ([j_{h_{A}^{α}}^{2} − \frac{1}{t} ∫_{0}^{1} |\frac{d}{dr} h_{A}^{α}(r)|^{2}dr] fη_{t})
= ∑_{A∈𝔨_{0}} μ_{t}^{0} ([j_{h_{A}^{α}}^{2} − \frac{1}{t} ∫_{0}^{1} |\frac{d}{dr} h_{A}^{α}(r)|^{2}dr] f) ,

where (by Eq. (5.8))

j_{h_{A}^{α}} = \frac{1}{\sqrt{α}t} ∫_{0}^{α}⟨A, db_{s}⟩ = \frac{1}{\sqrt{α}t} ∫_{0}^{α}⟨A, db_{s}⟩ = \frac{1}{\sqrt{α}t}⟨A, b_{α}⟩.

Using these facts and

∫_{0}^{1} |\frac{d}{dr} h_{A}^{α}(r)|^{2}dr = \frac{1}{α} ∫_{0}^{α} |A|^{2}dr = |A|^{2}.

we see that

J_{α}^{(2)} = \frac{1}{2}μ_{t}^{0} (\frac{1}{αt^{2}} |b_{α}|^{2} − \frac{1}{t} dim 𝔨)

and hence by Remark 2.17,

lim_{α→1} J_{α}^{(2)} = \frac{1}{2}μ_{t}^{0} (\frac{1}{t^{2}} |b_{1}|^{2} − \frac{1}{t} dim 𝔨) . (5.17)

Assembling Eqs. (5.16) and (5.17) shows that

lim_{α→1} J_{α} = \frac{1}{2}μ_{t}^{0} (Δ_{L(K)}f + [\frac{1}{t^{2}} |b_{1}|^{2} − \frac{1}{t} dim 𝔨] f) . (5.18)

Combining Eqs. (5.10), (5.12) and (5.18) proves the Theorem. Corollary 5.7. Suppose that u : L(K) → ℝ is a smooth cylinder function and let G(t, σ) = ∫_{L(K)} u(σγ^{−1})dμ_{t}^{0}(γ), (5.19) then

∂_{t}G(t, σ) = \frac{1}{2}△_{L(K)}G(t, σ) + ∫_{L(K)} V_{t}(γ)u(σγ^{−1})dμ_{t}^{0}(γ). (5.20)

Proof. As in the proof of Corollary 3.11, let u_{σ} : L(K) → ℝ be the cylinder function defined by u_{σ}(γ) = u(σγ^{−1}). By the Airault Malliavin Theorem 2.18,

∂_{t}G(t, σ) = ∫_{L(K)} [\frac{1}{2} (△_{L(K)}u_{σ}) (γ) + V_{t}(γ)u_{σ}(γ)] dμ_{t}^{0}(γ).

Using the same method of proof used for Corollary 3.11, we see that this equation is the same as Eq. (5.20).

6. Absolute continuity of heat kernel with respect to pinned Wiener measure

In this section we will prove the main Theorem 2.16. We will first need a couple of preliminary results.

Lemma 6.1 (Asymptotic properties of heat Kernels on K). The heat kernel, p_{t}^{K}, on K has the following properties:

1. lim_{t→0}(2πt)^{\frac{1}{2} dim 𝔨}p_{t}^{K} (e) = 1.

2. For every T < ∞, there is a constant M_{T} < ∞ such that

p_{t}^{K}(x) ≤ M_{T} t^{− dim 𝔨/2}e^{− \frac{1}{4t} d^{2}(e,x)} for all x ∈ K and 0 < t ≤ T

where d(x, y) is the distance associated to the bi-invariant Riemannian metric on K which agrees with ⟨·, ·⟩_{𝔨} at e ∈ K.

Proof. These are standard properties of heat kernels. For item 1., see Theorem 2.30 of [5]. See also [25]. For the second item see, for example, Theorem IX.1.2 in [32]. To apply this theorem, use the fact that K is compact so the modular function is constant. It is also necessary to note that the time parameter in [32] is twice our time parameter t.

Lemma 6.2 (μ_{t}^{0} → δ_{e} as t → 0). Let f : L(K) → ℝ be a continuous cylinder function, then

lim_{t→0+} μ_{t}^{0}(f) = f(e), (6.1)

where e denotes the identity loop in L(K), see Notation 1.2.

Proof. This result can be proved in a number of ways. For example one could use the Kolmogorov’s continuity criteria to show that μ_{t}^{0} concentrates near the identity loop as t → 0. See the argument in the proof of Item 1 of Theorem 2.3 in [12]. Rather than carry this out in full detail, we will only prove what we need.

Let ℙ be a partition of [0, 1] as in Eq. (2.3), f = F ∘ π_{ℙ} and ρ^{ℙ} : (0, ∞) × K^{n} → (0, ∞) be as in Eq. (2.10). By Lemma 6.1, there is a constant M < ∞ such that

ρ^{ℙ}(t, x) ≤ Mt^{\frac{1}{2} dim 𝔨} ∏_{i=1}^{n+1}(tΔ_{i}s)^{− dim 𝔨/2} exp (−\frac{1}{4tΔ_{i}s}d^{2}(e, x_{i−1}^{−1}x_{i})) for all t ∈ (0, 1],

where x = (x_{1}, ⋯, x_{n}), Δ_{i}s = s_{i} − s_{i−1}, and x_{0} = x_{n+1} = e ∈ K. By the left invariance of the Riemannian metric on K, d(x, y) = d(e, x^{−1}y), so the previous inequality may be written as

ρ^{ℙ}(t, x) ≤ M_{ℙ}t^{− \frac{n}{2} dim 𝔨} exp (−\frac{1}{4t} ∑_{i=1}^{n+1} \frac{d^{2}(x_{i−1}, x_{i})}{Δ_{i}s}) (6.2)

where M_{ℙ} := M ∏_{i=1}^{n+1}(Δ_{i}s)^{− dim 𝔨/2}. Now let δ > 0 be given, and suppose that d(e, x_{i}) ≥ δ for some i ∈ {1, 2, …, n}, then by the triangle inequality and the Cauchy-Schwarz inequality,

δ^{2} ≤ d^{2}(e, x_{i}) ≤ (∑_{j=1}^{i} d(x_{j−1}, x_{j}))^{2}

≤ (∑_{i=1}^{n+1} \frac{d(x_{j−1}, x_{j})}{\sqrt{Δ_{j}s}}\sqrt{Δ_{j}s})^{2} ≤ ∑_{i=1}^{n+1} Δ_{i}s · ∑_{i=1}^{n+1} \frac{d^{2}(x_{i−1}, x_{i})}{Δ_{i}s}
= ∑_{i=1}^{n+1} \frac{d^{2}(x_{i−1}, x_{i})}{Δ_{i}s}

Combining this estimate with Eq. (6.2) implies

ρ^{ℙ}(t, x) ≤ M_{ℙ}t^{− \frac{n}{2} dim 𝔨} exp (−\frac{1}{4t} |x|^{2}) (6.3)

where

|x| := max{d(e, x_{i}) : i = 1, 2, …, d}.

Therefore ρ^{ℙ}(t, ·) satisfies:

1. ρ^{ℙ}(t, x) > 0.

2. ∫_{K^{n}} ρ^{ℙ}(t, x)dx = 1 where dx is Haar measure on K^{n}.

3. For any δ > 0, ρ^{ℙ}(t, x) → 0 uniformly in x ∈ K^{n} with |x| ≥ δ.

It is now routine to show, using these three properties, that

lim_{t→0} ∫_{K^{n}} F (x)ρ^{ℙ}(t, x)dx = F (e, e, …, e)

which is equivalent to Eq. (6.1).

6.1. Proof of Theorem 2.16.

Proof. Let u be a smooth non-negative cylinder function on L (K) and let C_{t} be as in Eq. (2.16). Notice that \frac{d}{dt}C_{t} = c_{t} (c_{t} is defined in Eq. (2.19) of the Airault – Malliavin Theorem 2.18) and because of Lemma 6.1, lim_{t→0} C_{t} = 0. Define

H(t, σ) = ∫_{L(K)} u(σγ^{−1})dν_{t}^{0}(γ), and
F (t, σ) = e^{C_{t}} ∫_{L(K)} u (σγ^{−1}) dμ_{t}^{0} (γ) ,

then by Corollary 3.11

∂_{t}H(t, σ) = \frac{1}{2}△_{L(K)}H(t, σ) and lim_{t→0} H(t, σ) = u(σ) (6.4)

and by Corollary 5.7

∂_{t}F (t, σ) = \frac{1}{2}△_{L(K)}F (t, σ) + e^{C_{t}} ∫_{L(K)} (V_{t}(γ) + c_{t}) u(σγ^{−1})dμ_{t}^{0}(γ)
= \frac{1}{2}△_{L(K)}F (t, σ) + e^{C_{t}} ∫_{L(K)} \frac{1}{2t^{2}} |b_{1}(γ)|_{𝔨}^{2} u(σγ^{−1})dμ_{t}^{0}(γ)
≥ \frac{1}{2}△_{L(K)}F (t, σ).

Combining this with Lemma 6.2, shows that

∂_{t}F (t, σ) ≥ \frac{1}{2}△_{L(K)}F (t, σ) and lim_{t→0} F (t, σ) = u(σ). (6.5)

The idea now is to use Eqs. (6.4), (6.5) and the maximum principle to conclude that

(6.6) F (t, σ) ≥ H(t, σ) for all 0 ≤ t < ∞ and σ ∈ L(K).

We will postpone the full justification of Eq. (6.6) to Lemma 6.3 below. Writing out Eq. (6.6) when σ is the constant loop e, shows that

∫_{L(K)} u(γ^{−1})dν_{t}^{0}(γ) ≤ e^{C_{t}} ∫_{L(K)} u (γ^{−1}) dμ_{t}^{0} (γ)

for all non-negative smooth cylinder functions u. Replacing u by the cylinder function ũ(γ) = u(γ^{−1}) then implies that

∫_{L(K)} u(γ)dν_{t}^{0}(γ) ≤ e^{C_{t}} ∫_{L(K)} u (γ) dμ_{t}^{0} (γ) (6.7)

for all non-negative smooth cylinder functions u.

Since, by Lemma 2.7, bounded smooth cylinder functions are dense in

L^{2}(L(K), F, μ_{t}^{0} + ν_{t}^{0}),

by passing to the limit, we may conclude that Eq. (6.7) is valid for all bounded non-negative F – measurable functions u. By taking u to be characteristic functions and using the Radon-Nikodym theorem, Eq. (6.7) implies that ν_{t}^{0} is absolutely continuous relative to μ_{t}^{0}. Letting Z_{t} := dν_{t}^{0}/dμ_{t}^{0} we may conclude from Eq. (6.7) that

∫_{L(K)} u · (Z_{t} − e^{C_{t}}) dμ_{t}^{0} ≤ 0

for all bounded measurable functions u and hence that Z_{t} − e^{C_{t}} ≤ 0.

Lemma 6.3. Keeping the same notation as above, Eq. (6.6) is valid.

Proof. In order to justify the use of the maximum principle to prove Eq. (6.6), write u = U ∘ π_{ℙ}, where ℙ is a partition as in Eq. (2.3) and U : K^{n} → [0, ∞) is a smooth function. Then

H(t, σ) = ∫_{L(K)} u(σγ^{−1})dν_{t}^{0}(γ) = ∫_{L(K)} U(π_{ℙ}(σ)π_{ℙ}(γ)^{−1})dν_{t}^{0}(γ)
(6.8)  = H_{ℙ}(t, π_{ℙ}(σ)),

where for x ∈ K^{n}

H_{ℙ}(t, x) = ∫_{L(K)} U(xπ_{ℙ}(γ)^{−1})dν_{t}^{0}(γ)
(6.9)  = ∫_{K^{n}} U(xy^{−1})p_{t}^{ℙ}(y)dy,

and p_{t}^{ℙ}(y)dy = Law(π_{ℙ}(Σ(t, ·)). By the proof of Proposition 3.9, π_{ℙ}(Σ(t, ·) is a diffusion on K^{n} with elliptic generator L_{ℙ}^{0} defined in Eq. (3.9). Thus p_{t}^{ℙ}(y) is the smooth heat kernel for the operator e^{tL_{ℙ}^{0}/2}. This shows that H_{ℙ}(t, x) is smooth on (0, ∞) × K^{n}. Using this information, Eq. (6.4) may be recast as the finite dimensional statement

∂_{t}H_{ℙ}(t, x) = \frac{1}{2}L_{ℙ}^{0}H_{ℙ}(t, x) and lim_{t→0} H_{ℙ}(t, x) = U(x). (6.10)

Similarly

F (t, σ) = e^{C_{t}} ∫_{L(K)} u (σγ^{−1}) dμ_{t}^{0} (γ) = e^{C_{t}} ∫_{L(K)} U(π_{ℙ}(σ)π_{ℙ}(γ)^{−1})dμ_{t}^{0} (γ)
= F_{ℙ}(t, π_{ℙ}(σ)),

where for x ∈ K^{n}

F_{ℙ}(t, x) = e^{C_{t}} ∫_{L(K)} U(xπ_{ℙ}(γ)^{−1})dμ_{t}^{0}(γ)
= e^{C_{t}} ∫_{K^{n}} U(xy^{−1})ρ^{ℙ}(t, y)dy

where ρ^{ℙ} : (0, ∞) × K^{n} → (0, ∞) is the smooth function defined in Eq. (2.10) of Definition 2.11. This shows that F_{ℙ}(t, x) is smooth on (0, ∞) × K^{n}. Using this information, Eq. (6.4) may be recast as the finite dimensional statement

∂_{t}F_{ℙ}(t, x) ≥ \frac{1}{2}L_{ℙ}^{0}F_{ℙ}(t, x) and lim_{t→0} F_{ℙ}(t, x) = U(x). (6.11)

Now there is no problem in applying the maximum principle on K^{n}, using Eqs. (6.10) and (6.11), to conclude that

F_{ℙ}(t, x) ≥ H_{ℙ}(t, x) for all 0 ≤ t < ∞ and x ∈ K^{n}.

This finishes the proof since this last assertion is equivalent (6.6).

7. The K = ℝ^{d} and S^{1} cases

In this section, we will work out the explicit relationship between μ_{t}^{0} and ν_{t}^{0} in the case that K is the abelian Lie group ℝ^{d} or S^{1}.

7.1. The K = ℝ^{d} case. Let K be the Lie group ℝ^{d} with group operation being addition. The Lie algebra of ℝ^{d} is 𝔨 =ℝ^{d} with the trivial lie bracket, [a, b] = 0 for all a, b ∈ ℝ^{d}. Although ℝ^{d} is not compact and is not being represented as a matrix group, the theory above easily extends to this case. There is one notational point to take care of now. Namely, the matrix expression of the form g^{−1}δg must now be interpreted as L_{g^{−1}∗}δg = δg. We will assume that ⟨a, b⟩ = a · b is the usual dot product, although any inner product would work.

Lemma 7.1. On the loop space of ℝ^{d}, L(ℝ^{d}), the heat kernel measures ν_{t}^{0} and the pinned Wiener measures, μ_{t}^{0}, are the same.

Proof. The process Σ^{0}(t, s) in Theorem 2.13 and the process g in Eq. (2.5) of Definition 2.8 are explicitly given by Σ^{0}(t, s) = χ(t, s) and g_{s} = β(t, s) respectively. Since g_{s} = β(t, s) is a standard Brownian motion with variance t, the pinned Wiener measure μ_{t}^{0} = Law(g_{·}|g_{1} = 0) is the law of an ℝ^{d} – valued Brownian bridge with variance t. But s → χ(t, s) is a Brownian bridge with variance t (see Remark 2.12), so that μ_{t}^{0} = Law(χ(t, ·)) = Law(Σ^{0}(t, ·)) = ν_{t}^{0}.

7.2. The K = S^{1} case. Let K = S^{1} = {z ∈ ℂ : |z| = 1}. The Lie algebra of K is 𝔨 = iℝ with the trivial Lie bracket. We will identify with 𝔨 = iℝ with ℝ, putting in the i explicitly when needed. Let ⟨a, b⟩ = ab for a, b ∈ ℝ ≅iℝ =𝔨.

Remark 7.2. Let p_{t}(x) = p_{t}^{ℝ}(x) = (2πt)^{−1/2} exp (−\frac{1}{2t}x^{2}) be the heat kernel on ℝ, and q_{t}(z) = \frac{1}{2π} p_{t}^{S^{1}}(z) denote the heat kernel on S^{1} relative to the un-normalized Haar measure “dθ,” i.e. for f : S^{1} → ℝ,

∫_{S^{1}} fdθ := ∫_{0}^{2π} f(e^{iθ})dθ.

The well know relationship between q_{t} and p_{t} is

q_{t}(e^{iθ}) = ∑_{n=−∞}^{∞} p_{t}(θ − 2πn) for θ ∈ ℝ. (7.1)

To check this, suppose that f_{0} : S^{1} → ℝ is a continuous function. Then

f(t, z) = ∫_{0}^{2π} f_{0}(ze^{−iα})q_{t}(e^{iα})dα

solves the heat equation on S^{1} which is equivalent to saying that F (t, θ) := f(t, e^{iθ}) solves the heat equation on ℝ. Since F is a bounded solution to the heat equation it is given by

∫_{0}^{2π} f_{0}(ze^{−iα})q_{t}(e^{iα})dα = F (t, θ) = ∫_{−∞}^{∞} F (0, θ − α)p_{t}(α)dα
= ∫_{−∞}^{∞} f_{0}(ze^{−iα})p_{t}(α)dα

= ∑_{n=−∞}^{∞} ∫_{0}^{2π} f_{0}(ze^{−i(α−2πn)})p_{t}(α − 2πn)dα

= ∑_{n=−∞}^{∞} ∫_{0}^{2π} f_{0}(ze^{−iα})p_{t}(α − 2πn)dα

where z = e^{iθ}. This equation, holding for all continuous f_{0} : S^{1} → ℝ, proves Eq. (7.1).

Definition 7.3. For n ∈ ℤ, let h_{n}(s) := 2πns,

z_{n}(s) = e^{i2πns} = e^{ih_{n}(s)}

and let ν_{t}^{n} be the left translation of ν_{t}^{0} by z_{n}, i.e. ν_{t}^{n} is the probability measure on L(S^{1}) such that

∫_{L(S^{1})} f(σ)dν_{t}^{n}(σ) = ∫_{L(S^{1})} f(z_{n}σ)dν_{t}^{0}(σ).

Also let L_{n}(S^{1}) denote those σ ∈ L(S^{1}) which are homotopic to z_{n}.

Remark 7.4. The loops {z_{n}}_{n=−∞}^{∞} are representatives from each of the homotopy classes of L(S^{1}), i.e. L(S^{1}) is the disjoint union of {L_{n}(S^{1})}_{n=−∞}^{∞} . By the construction of ν_{t}^{0} in Theorem 2.13, the measure ν_{t}^{0} is concentrated on L_{0}(S_{1}) and therefore ν_{t}^{n} is concentrated on L_{n}(S^{1}), i.e. ν_{t}^{n}(L_{m}(S_{1})) = δ_{mn}.

Proposition 7.5. The relationship between pinned Wiener measure μ_{t}^{0} and heat kernel measure ν_{t}^{0} on L(S^{1}) is

μ_{t}^{0} = \frac{1}{q_{t}(1)} ∑_{n=−∞}^{∞} p_{t}(2πn)ν_{t}^{n}.

In particular

μ_{t}^{0}|_{L_{0}(S^{1})} = \frac{1}{q_{t}(1)}p_{t}(0)ν_{t}^{0} = ( ∑_{n=−∞}^{∞} e^{− \frac{1}{2t} (2πn)^{2}})^{−1} ν_{t}^{0}. (7.2)

Proof. To simplify notation, let B_{s} = β(t, s). Using Itô’s formula, one easily shows that the process Σ^{0}(t, s) in Theorem 2.13 and the process g in Eq. (2.5) of Definition 2.8 are given by Σ^{0}(t, s) = e^{iχ(t,s)} and g_{s} = e^{iβ(t,s)} = e^{iB_{s}} respectively. Suppose that f : L(S^{1}) → ℝ is a cylinder function as in Definition 2.4. then for α ∈ (s_{n}, 1),

μ_{t}^{0}(f) = 𝔼 [f(g)\frac{q_{t(1−α)}(g_{α})}{q_{t}(1)}]

= 𝔼 [f(e^{iB_{·}})\frac{q_{t(1−α)}(e^{iB_{α}})}{q_{t}(1)}] = \frac{1}{q_{t}(1)}𝔼 [f(e^{iB_{·}}) ∑_{n=−∞}^{∞} p_{t(1−α)}(B_{α} − 2πn)]

(7.3) = \frac{1}{q_{t}(1)} ∑_{n=−∞}^{∞} 𝔼 [f(e^{iB_{·}})p_{t(1−α)}(B_{α} − 2πn)] .

Let h_{n}(s) := 2πns, h_{n}^{α}(s) = 2πn (s ∧ α) , and F (B) = f(e^{iB_{·}}), so that F is a bounded cylinder function W (ℝ). By the Cameron-Martin theorem (making the translation B_{·} → B_{·} + h_{n}^{α}),

𝔼[F (B)p_{t(1−α)}(B_{α} − 2πn)]

= 𝔼 [F (B + h_{n}^{α})p_{t(1−α)}(B_{α} − 2πn(1 − α)) exp (−\frac{1}{t} ∫_{0}^{α} 2πn dB_{s} − \frac{1}{2t} ∫_{0}^{α}(2πn)^{2}ds)]

= 𝔼 [F (B + h_{n})p_{t(1−α)}(B_{α} − 2πn(1 − α)) exp (−\frac{2πn}{t}B_{α} − \frac{1}{2t}α(2πn)^{2})] . (7.4)

By direct computation,

p_{t(1−α)}(x − y(1 − α)) = p_{t(1−α)}(x) · exp (\frac{1}{t}xy − \frac{1}{2t}(1 − α)y^{2})

and thus taking x = B_{α} and y = 2πn,

p_{t(1−α)}(B_{α} − 2πn(1 − α)) · exp (−\frac{2πn}{t}B_{α} − \frac{1}{2t}α(2πn)^{2})

= p_{t(1−α)}(B_{α}) · exp (−\frac{1}{2t}(2πn)^{2}) . (7.5)

Combining Eqs. (7.4) and (7.5) shows that

𝔼[F (B)p_{t(1−α)}(B_{α} − 2πn)]

= (2πt)^{−1/2} exp (−\frac{1}{2t}(2πn)^{2}) 𝔼 [F (B + h_{n})\frac{p_{t(1−α)}(B_{α})}{p_{t}(0)}]

= p_{t}(2πn)𝔼 [F (χ(t, ·) + h_{n})] wherein the second equality we have used Eq. (2.11) of Remark 2.12. Using this equation, with F (B) = f(e^{iB}), in Eq. (7.3) gives

μ_{t}^{0}(f) = \frac{1}{q_{t}(1)} ∑_{n=−∞}^{∞} p_{t}(2πn)𝔼 [f(e^{i(χ(t,·)+h_{n})})]

= \frac{1}{q_{t}(1)} ∑_{n=−∞}^{∞} p_{t}(2πn)𝔼 [f(z_{n}e^{iχ(t,·)})]

= \frac{1}{q_{t}(1)} ∑_{n=−∞}^{∞} p_{t}(2πn)ν_{t}^{n}(f).

8. Appendix (Quadratic variations)

Lemma 8.1. As above, for A ∈ 𝔨 let β^{A}(t, s) = ⟨β(t, s), A⟩_{𝔨} and χ^{A}(t, s) = ⟨χ(t, s), A⟩_{𝔨} . Let A, B ∈ 𝔨 and s, σ ∈ [0, 1], then

β^{A}(dt, s)β^{B}(dt, σ) = ⟨A, B⟩_{𝔨} G (s, σ) dt,
χ^{A}(dt, s)χ^{B}(dt, σ) = ⟨A, B⟩_{𝔨} G_{0} (s, σ) dt

and for t, τ ∈ [0, ∞),

β^{A}(t, ds)β^{B}(τ, ds) = ⟨A, B⟩_{𝔨} G(t, τ)ds.

Proof. Let {𝔊_{t}} be an abstract filtration (satisfying the “usual hypothesis”) and suppose that M_{t} and N_{t} are two continuous {𝔊_{t}} adapted processes such that (M_{t} − M_{s}, N_{t} − N_{s}) is independent of 𝔊_{s} for all t > s and 𝔼M_{t} = 𝔼N_{t} = 0 for all t ≥ 0. Then clearly M and N are {𝔊_{t}} – martingales. We now also assert that

(8.1) M_{t}N_{t} − 𝔼 [M_{t}N_{t}] is a martingale

Assuming Eq. (8.1) for the moment, we may conclude the differential M_{dt}N_{dt} of the quadratic co-variation of M and N is given by

M_{dt}N_{dt} = d_{t}𝔼[M_{t}N_{t}]. (8.2)

The lemma then follows from repeated application of Eq. (8.2). For example, taking M_{t} = β^{A}(t, s) and N_{t} = β^{B}(t, σ), we learn that

β^{A}(dt, s)β^{B}(dt, σ) = d_{t}𝔼 [β^{A}(t, s)β^{B}(t, σ)] = ⟨A, B⟩_{𝔨} G (s, σ) dt.

To prove Eq. (8.1), let t > s, ΔM = M_{t} −M_{s}, ΔN = N_{t} −N_{s} and 𝔼_{s} = 𝔼(·|𝔊_{s}). Then using the martingale properties of M and N and the independent increment assumption we find

𝔼_{s} [M_{t}N_{t} − M_{s}N_{s}] = 𝔼_{s} [(M_{s} + ΔM)(N_{s} + ΔN) − M_{s}N_{s}] = 𝔼_{s} [ΔMΔN]
= 𝔼[(M_{t} − M_{s}) (N_{t} − N_{s})] = 𝔼[(M_{t} − M_{s}) (N_{t} + N_{s})]
= 𝔼[M_{t}N_{t}] − 𝔼[M_{s}N_{s}].

Rearranging the terms of the result of this computation shows that

𝔼_{s} [M_{t}N_{t} − 𝔼[M_{t}N_{t}]] = M_{s}N_{s} − 𝔼[M_{s}N_{s}]

as desired.

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Department of Mathematics, 0112, University of California, San Diego, La Jolla, CA 92093-0112

E-mail address: driver@euclid.ucsd.edu

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