Itô maps and analysis on path spaces

DOI10.1007/S00209-007-0145-0zbMATH Open1181.60078arXivmath/0605089OpenAlexW2116714817MaRDI QIDQ2384754

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Publication date: 10 October 2007

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Abstract: We consider versions of Malliavin calculus on path spaces of compact manifolds with diffusion measures, defining Gross-Sobolev spaces of differentiable functions and proving their intertwining with solution maps, I, of certain stochastic differential equations. This is shown to shed light on fundamental uniqueness questions for this calculus including uniqueness of the closed derivative operator $d$ and Markov uniqueness of the associated Dirichlet form. A continuity result for the divergence operator by Kree and Kree is extended to this situation. The regularity of conditional expectations of smooth functionals of classical Wiener space, given I, is considered and shown to have strong implications for these questions. A major role is played by the (possibly sub-Riemannian) connections induced by stochastic differential equations: Damped Markovian connections are used for the covariant derivatives.

Full work available at URL: https://arxiv.org/abs/math/0605089

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