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Logarithmic Sobolev inequalities on path spaces over Riemannian manifolds. (English) Zbl 0892.58083

Summary: Let \(W_o(M)\) be the space of paths of unit time length on a connected, complete Riemannian manifold \(M\) such that \(\gamma (0)=o\), a fixed point on \(M\), and \(\nu\) the Wiener measure on \(W_o(M)\) (the law of Brownian motion on \(M\) starting at \(o)\). If the Ricci curvature is bounded by \(c\), then the following logarithmic Sobolev inequality holds: \[ \int_{W_o(M)} F^2\log| F| d\nu\leq e^{3c} \| DF\|^2 +\| F\|^2 \log \| F\|. \]

MSC:

58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
58J65 Diffusion processes and stochastic analysis on manifolds
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