A Cameron-Martin type quasi-invariance theorem for Brownian motion on a compact Riemannian manifold

Abstract

Let M be a compact manifold without boundary, o be a fixed base point in M, g be a Riemannian metric on M, and ▽ be a g-compatible covariant derivative on TM—the tangent space to M. Assume the torsion ( T) of ▽ satisfies the skew symmetry condition: g〈 T〈 X, Y〉, Y〉 ≡ 0 for all vector fields X and Y on M. (For example, take ▽ to be the Levi-Civita covariant derivative on ( M, g).) Also let ν denote the Wiener measure on W 0( M) = { ω ϵ C([0, 1], M): ω(0) = o}, and let H( ω)( s) denote stochastic parallel translation (relative to ν) along the path ω ϵ W 0( M) up to time s ϵ [0, 1]. Given a C 1-function h: [0, 1] → T 0 M, it is shown that the differential equation σ( t) = H( σ( t)) h with initial condition σ(0) = id: W( M) → W( M) has a solution σ: R → Maps(W(M), W(M)) —the measurable maps from W( M) to W( M). This function (σ) is a flow on W( M), i.e., for all t, τϵ R , σ(t+τ)=σ(t)σ(τ) . Furthermore σ( t) has the quasi-invariance property: the law (σ(t) ∗ν ) of σ( t) with respect to the Wiener measure (ν) is equivalent to ν for all tϵ. R . This result is used to prove an integration by parts formula for the h-derivative δ h f defined by ∂ hf(ω) ≡ (d dt) ¦ 0f(σ(t)(ω)) , where f is a “ C 2-cylinder” function on W( M).