A class of integration by parts formulae in stochastic analysis I

  • K. D. Elworthy
  • Xue-Mei Li


Consider a Stratonovich stochastic differential equation
$$d{{\chi }_{t}}=X\left( {{\chi }_{_{t}}} \right)od{{B}_{t}}+A\left( {{\chi }_{t}} \right)dt$$
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
$$Edf\left( T\xi .\left( h. \right) \right)=EF\left( \xi .\left( \chi \right) \right)\int_{0}^{T}{\left\langle T{{\xi }_{s}} \right.}\left( {{{\dot{h}}}_{s}} \right),X\left( {{\xi }_{s}}\left( \chi \right) \right)d\left. {{B}_{s}} \right\rangle $$
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 ≤ st} Taking conditional expectation with respect to.F T x , formula (1.2) yields integration by parts formulae on C x (M) of the form
$$EdF\left( \gamma \right)\left( {{\overline{V}}^{h}} \right)=EF\left( \gamma \right)\delta {{\overline{V}}^{h}}\left( \gamma \right)$$
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
$$\delta {{\overline{V}}^{h}}(\gamma )=IE\left\{ \int_{0}^{T}{<T{{\xi }_{s}}({{{\dot{h}}}_{s}}),X({{\xi }_{s}}(x))d{{B}_{s}}>|\xi .(x)=\gamma } \right\}$$


Vector Field Stochastic Differential Equation Solution Flow Stochastic Analysis Compact Riemannian Manifold 


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Copyright information

© Springer-Verlag Tokyo 1996

Authors and Affiliations

  • K. D. Elworthy
    • 1
  • Xue-Mei Li
    • 1
  1. 1.Mathematics InstituteUniversity of WarwickCoventryUK

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