Equivalence of heat kernel measure and pinned Wiener measure on loop groups

Shigeki Aida^{†} and Bruce K. Driver^{∗}

September 6, 2000 File:equivfinb.tex, Ref (*) : I 360 - 2000360

Abstract:

We show that heat kernel measure and pinned Wiener measure on loop groups over simply connected compact Lie groups are equivalent.

1 French Abridged Version

Abstract:

We show that heat kernel measure and pinned Wiener measure on loop groups over simply connected compact Lie groups are equivalent.

Let K be a connected compact Lie group, e ∈ K be the identity, 𝔨 ≡ T_{e}K be the Lie algebra of K, and ⟨·, ·⟩ be an Ad_{K}-invariant inner product on 𝔨. Also let L be the loop group in Eq. (2.1), μ_{t}^{0} be pinned Wiener measure on L with variance t and ν_{t}^{0} be the time t heat kernel measure on L, so that ν_{t}^{0} is the law of the process s → Σ(t, s) described in Eq. (2.4). In [6], it was shown that the heat kernel measure ν_{t}^{0} is absolute continuous relative to the pinned Wiener measure μ_{t}^{0} and it was conjectured that these two measures are equivalent if K is simply connected. In this note, we will prove in Theorem 3.2 that this conjecture is true. The idea of the proof is as follows.

Let Z_{t}(γ) = \frac{dν_{t}^{0}}{dμ_{t}^{0}} (γ), N := \{γ ∈ L : Z_{t}(γ) = 0\}, and H^{1}(K) denotes the subgroup of finite energy loops in L. Using the quasi-invariance of the pinned measure μ_{t}^{0} proved in Malliavin and Malliavin [9] and of ν_{t}^{0} proved in Driver [4] under left translation by H^{1}(K), one shows that h·N = N modulo sets of μ_{t}^{0} – measure zero for all h ∈ H^{1}(K). Using Gross’ ergodicity theorem, see [8] and Aida [1], it follows that either μ_{t}^{0}(N) = 0 or that μ_{t}^{0}(L \ N) = 0. Since ν_{t}^{0} is not the zero measure, we conclude that μ_{t}^{0}(N) = 0, i.e. that Z(γ) > 0 on L for μ_{t}^{0} - a.e. γ. That is to say that μ_{t}^{0} and ν_{t}^{0} are equivalent. A generalization of this result to non-simply connected groups K is given in Corollary 3.3.

2 Introduction

Let K be a connected compact Lie group, e ∈ K be the identity, 𝔨 ≡ T_{e}K be the Lie algebra of K and ⟨·, ·⟩ be an Ad_{K}-invariant inner product on 𝔨. (To simplify notation later we will, with out loss of generality, assume that K is a matrix group.) The loop group on K is defined by

L = L (K) ≡ {γ ∈ C([0, 1], K) : γ(0) = γ(1) = e}  (2.1)

and when K is not simply connected, we let L_{0} denote those loops in L which are homotopic to the constant loop at e ∈ K. We view L as a topological space equipped with the topology of uniform convergence.

Let μ_{t}^{0} denote pinned Wiener measure with variance t. This measure may be described as the law of the process {g_{s} : 0 ≤ s ≤ 1} pinned so that g_{1} = e, where g is the solution to the Stratonovich stochastic differential equation

dg_{s} = g_{s} ∘ \sqrt{t}db_{s} with g_{0} = e.  (2.2)

Here b is a standard 𝔨 – valued Brownian motion with covariance determined by ⟨·, ·⟩. As introduced in [5] and [10], the heat kernel measure ν_{t}^{0} on L is the time t distribution of a certain L – valued Brownian motion which we now describe.

Let {χ(t, s)}_{0≤s≤1,0≤t<∞} be a 𝔨 – valued Brownian bridge sheet, i.e. χ is a 𝔨 – valued centered continuous Gaussian process such that

𝔼[⟨A, χ(t, s)⟩⟨B, χ(τ, σ)⟩] = ⟨A, B⟩ min(t, τ) (min(s, σ) − σs)  (2.3)

for all s, σ ∈ [0, 1], t, τ ∈ [0, ∞) and A, B ∈ 𝔨. Let Σ(t, s) for 0 ≤ t < ∞ and 0 ≤ s ≤ 1 denote the solution to the Stratonovich stochastic differential equation in t (with s as a parameter),

Σ(dt, s) = Σ(t, s) ∘ χ(dt, s) with Σ(0, s) = e ∀s ∈ [0, 1].  (2.4)

By Malliavin [10] (see also Baxendale [3] or Driver [4]), we may and do choose Σ(t, s) to be jointly continuous in (t, s). The heat kernel measure ν_{t}^{0} at time t > 0 is the law of the process s → Σ(t, s) ∈ K. Notice by construction that ν_{t}^{0}(L_{0}) = 1. For further details and motivations for these constructions the reader is referred to [4, 5, 6].

In [6], it was shown that ν_{t}^{0} is absolute continuous relative to μ_{t}^{0} and it was conjectured that these two measures are equivalent if K is simply connected. In this note, we will prove in Theorem 3.2 that this conjecture is true. The proof is based on an elementary fact (Lemma 3.1 below) on the equivalence of two quasi-invariant measures.

3 An elementary lemma and the equivalence of ν_{t}^{0} and μ_{t}^{0}

Let X be a topological group and Y be a subgroup. For a finite Borel measure ν and h ∈ Y, define the new measure ν^{h} on X by ν^{h}(A) = ν(h · A), where A is a Borel measurable subset and h · A = {hx | x ∈ A}.

We recall the basic notion of measures on groups. We say that ν is quasi-invariant relative to Y if ν^{h} is equivalent to ν for all h ∈ Y. For a quasi-invariant measure ν relative to Y, ν is said to be ergodic relative to the left action of Y if a Borel measurable subset A satisfying ν((A \ h · A) ∪ (h · A \ A)) = 0 for any h ∈ Y, implies that either ν(X \ A) = 0 or ν(A) = 0.

Lemma 3.1 Let ν_{1} and ν_{2} be quasi-invariant finite non-zero measures on X relative to a subgroup Y. Let α_{i}^{h}(x) = \frac{dν_{i}^{h}}{dν_{i}} (x) (i = 1, 2). Suppose that (1) ν_{1} is ergodic relative to Y and that

(2) ν_{2} is absolute continuous relative to ν_{1}.

Then ν_{1} and ν_{2} are equivalent.

Proof. Let Z(x) = \frac{dν_{2}}{dν_{1}} (x). We need only prove that Z(x) > 0 for ν_{1}-a.e. x. We will do this by showing that ν_{1}(N) = ν_{1}({x ∈ X | Z(x) = 0}) = 0.

By definition of α_{i}^{h}(x),

∫_{X} f(h^{−1}x)dν_{i}(x) = ∫_{X} f(x)α_{i}^{h}(x)dν_{i}(x) (i = 1, 2)  (3.1)

for all bounded measurable functions f on X and h ∈ Y. Using Eq. (3.1) and the definition of Z we have

∫_{X} f(h^{−1}x)dν_{2}(x) = ∫_{X} f(h^{−1}x)Z(x)dν_{1}(x) = ∫_{X} f(x)Z(hx)α_{1}^{h}(x)dν_{1}(x)  (3.2)

and

∫_{X} f(h^{−1}x)dν_{2}(x) = ∫_{X} f(x)α_{2}^{h}(x)dν_{2}(x) = ∫_{X} f(x)α_{2}^{h}(x)Z(x)dν_{1}(x).  (3.3)

Since these equations hold for all bounded measurable functions, it follows

Z(hx)α_{1}^{h}(x) = Z(x)α_{2}^{h}(x) for ν_{1} – a.e. x  (3.4)

for all h ∈ Y. Similar calculations shows that α_{i}^{h^{−1}}(hx)α_{i}^{h}(x) = 1 for ν_{i} – a.e. x, i = 1, 2 and h ∈ Y. In particular

α_{i}^{h}(x) > 0 for ν_{i} – a.e. x.  (3.5)

Thus we may write Eq. (3.4) as

Z(hx) = Z(x)\frac{α_{2}^{h}(x)}{α_{1}^{h}(x)}.  (3.6)

By Eqs. (3.5) and (3.6) and the fact that ν_{2}(N) = 0,

h · N = N up to a ν_{1}-null set  (3.7)

for all h ∈ Y. Hence by the ergodicity of ν_{1}, either ν_{1}(X \ N) = 0 or ν_{1}(N) = 0. Since ν_{2} is not zero we know ν_{1}(X \ N) > 0 and therefore, ν_{1}(N) = 0 which completes the proof.

Now we are going to prove the equivalence of heat kernel measure and pinned Wiener measure. Let μ̃_{t}^{0} be the restriction of the pinned Wiener measure μ_{t}^{0} to L_{0}. The absolute continuity of ν_{t}^{0} with μ_{t}^{0} was proved in [6]. Let Z_{t}(γ) = \frac{dν_{t}^{0}}{dμ_{t}^{0}} (γ).

Theorem 3.2 The Radon – Nikodým derivative Z_{t}(γ) is positive for μ̃_{t}^{0}-a.e. γ. Namely μ_{t}^{0} and ν_{t}^{0} are equivalent on L_{0}

Proof. Let H^{1}(K) and H^{1}(K)_{0} denote the subgroups of finite energy loops in L and L_{0} respectively. The quasi-invariance of the un-pinned measure relative to H^{1}(K) is standard (see [2], [7], [11]). However the quasi-invariance of pinned Wiener measure relative to H^{1}(K) is much more delicate and is due to M.-P. Malliavin and P. Malliavin [9]. The quasi-invariance of heat kernel measure ν_{t}^{0} relative to H^{1}(K)_{0} was proved in [4]. Furthermore the ergodicity of μ̃_{t}^{0} relative to H^{1}(K)_{0} was proved by Gross [8], see also Aida [1] for this result and its generalizations. Hence the theorem is a consequence of Lemma 3.1 after setting X = L_{0}, Y = H^{1}(K)_{0}, ν_{1} = μ̃_{t}^{0} and ν_{2} = ν_{t}^{0}.

Corollary 3.3 For each h ∈ H^{1}(K), let ν_{t}^{0}(h, A) := ν_{t}^{0}(h^{−1}A) – heat kernel measure on L starting at h. Let Π ⊂ H^{1}(K) be chosen so that to each homotopy class in L, there is a unique representative in Π. Then μ_{t}^{0} and ∑_{h∈Π} ν_{t}^{0}(h, ·) are equivalent measures on L.

Proof. For h ∈ H^{1}(K), let L_{h} denote those loops in L which are homotopic to h. By Theorem 3.2, ν_{t}^{0}(h, ·) is equivalent to μ̃_{t}^{0}(h^{−1}(·)) on L_{h^{−1}}. By the quasi-invariance of μ_{t}^{0} relative to H^{1}(K), it follows that μ̃_{t}^{0}(h^{−1}(·)) is equivalent to μ_{t}^{0}|_{L_{h^{−1}}} on L_{h^{−1}}. Therefore ν_{t}^{0}(h, ·) is equivalent to μ_{t}^{0}|_{L_{h^{−1}}} on L_{h^{−1}} and hence ∑_{h∈Π} ν_{t}^{0}(h, ·) is equivalent to ∑_{h∈Π} μ_{t}^{0}|_{L_{h^{−1}}} = μ_{t}^{0}.

†This research was partially supported by Grant-in-Aid for Scientific Research (C) No. 12640173. ∗This research was partially supported by NSF Grant DMS 99-71036.

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S. Aida Department of Mathematical Science, Graduate School of Engineering Science Osaka University, Toyonaka, 560-8531, JAPAN e-mail: aida@sigmath.es.osaka-u.ac.jp

Bruce K. Driver Department of Mathematics, 0112 University of California at San Diego La Jolla, CA, 92093-0112, USA e-mail: driver@euclid.ucsd.edu