Integration by parts formulae for degenerate diffusion measures on path spaces and diffeomorphism groups

Abstract

Integration by parts formulae are given for a class of measures on the space of paths of a smooth manifold $M$ determined by the laws of degenerate diffusions. The mother of such formulae, on the path space of diffeomorphism group of $M$ is shown to arise from a quasi-invariance property of measures determined by stochastic flows. From this the other formulae are derived by filtering out redundant noise using an associated LeJan-Watanabe connection.

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• Ito maps and analysis on path spaces
• Semidirect products of groups of loops and groups of diffeomorphisms of real, complex and quaternion manifolds