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The abstract Riemannian path space. (English) Zbl 0949.60064

Summary: On the Wiener space \(\Omega\), we introduce an abstract Ricci process \(A_t\) and a pseudo-gradient \(F\rightarrow{F}^\sharp\) which are compatible through an integration by parts formula. They give rise to a \(\sharp\)-Sobolev space on \(\Omega\), logarithmic Sobolev inequalities, and capacities, which are tight on Hölder compact sets of \(\Omega\). These are then applied to the path space over a Riemannian manifold.

MSC:

60H07 Stochastic calculus of variations and the Malliavin calculus
58D20 Measures (Gaussian, cylindrical, etc.) on manifolds of maps
58J99 Partial differential equations on manifolds; differential operators
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H25 Random operators and equations (aspects of stochastic analysis)
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