A class of integration by parts formulae in stochastic analysis I

Consider a Stratonovich stochastic differential equation 1.1 $$d{{\chi }_{t}}=X\left( {{\chi }_{_{t}}} \right)od{{B}_{t}}+A\left( {{\chi }_{t}} \right)dt$$ with C∞ coefficients on a compact Riemannian manifold M, with associated differential generator $$A=\frac{1}{2}{{\Delta }_{M}}+Z$$ and solution flow {ξt : t ≥ 0} of random smooth diffeomorphisms of M. Let Tξt: TM → TM be the induced map on the tangent bundle of M obtained by differentiating ξt with respect to the initial point. Following an observation by A. Thalmaier we extend the basic formula of [EL94] to obtain 1.2 $$Edf\left( T\xi .\left( h. \right) \right)=EF\left( \xi .\left( \chi \right) \right)\int_{0}^{T}{\left\langle T{{\xi }_{s}} \right.}\left( {{{\dot{h}}}_{s}} \right),X\left( {{\xi }_{s}}\left( \chi \right) \right)d\left. {{B}_{s}} \right\rangle $$ where $$F\in FC_{b}^{\infty }\left( {{C}_{\chi }}\left( M \right) \right)$$ , the space of smooth cylindrical functions on the space C x (M) of continuous paths γ : [0,T] → M with γ(0) = x, dF is its derivative, and h. is a suitable adapted process with sample paths in the Cameron-Martin space L 0 2,1 ([0,T];T x M).Set F t x = σ{ξs(x) : 0 ≤ s ≤ t} Taking conditional expectation with respect to.F T x , formula (1.2) yields integration by parts formulae on C x (M) of the form 1.3 $$EdF\left( \gamma \right)\left( {{\overline{V}}^{h}} \right)=EF\left( \gamma \right)\delta {{\overline{V}}^{h}}\left( \gamma \right)$$ where $${{\overline{V}}^{h}}$$ is the vector field on C x(M) $${{\overline{V}}^{h}}{{\left( \gamma \right)}_{t}}-E\left\{ T{{\xi }_{t}}\left( {{h}_{t}} \right)\left| \xi .\left( \chi \right)=\gamma \right. \right\}$$ and $$\delta {{\overline{V}}^{h}}:{{C}_{x}}\left( M \right)\to $$ is given by $$\delta {{\overline{V}}^{h}}(\gamma )=IE\left\{ \int_{0}^{T}{ |\xi .(x)=\gamma } \right\}$$ .