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

AbstractLet 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 W0(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 W0(M) up to time s ϵ [0, 1]. Given a C1-function h: [0, 1] → T0M, 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, τϵR, 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 δhf defined by ∂hf(ω) ≡ (ddt)¦0f(σ(t)(ω)), where f is a “C2-cylinder” function on W(M)