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Differential calculus on path and loop spaces. I: Logarithmic Sobolev inequalities on path spaces. (English. Abridged French version) Zbl 0837.60053

The authors prove a logarithmic Sobolev inequality (L.S.I.) on a path space over a compact Riemannian manifold. Path space means a space of continuous paths over the manifold whose starting point is fixed and the probability measure is given by the Brownian motion measure. The proof is similar to Gross’ one in the case of compact Lie group. More precisely, they embed the manifold into a Euclidean space isometrically and consider the solution of an S.D.E. which is called a gradient Brownian system. Using this, one can pull back a function on the path space to a function on the Wiener space preserving the law. The composition function satisfies an L.S.I. because of the Gross original L.S.I. on Wiener space. The key of the proof is the computation of a certain conditional expectation of the Malliavin derivative of the composition function by using Elworthy and Tor’s observation.
Reviewer: S.Aida (Sendai)

MSC:

60H07 Stochastic calculus of variations and the Malliavin calculus
58J65 Diffusion processes and stochastic analysis on manifolds
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