Martingale Representation and a Simple Proof of Logarithmic Sobolev Inequalities on Path Spaces
Mireille Capitaine, Universite Paul-Sabatier
Elton P. Hsu, Northwestern University
Michel Ledoux, Universite Paul-Sabatier
Abstract
We show how the Clark-Ocone-Haussmann formula for
Brownian motion on a compact Riemannian manifold put forward by
S. Fang in his proof of the spectral gap inequality for the
Ornstein-Uhlenbeck operator on the path space can yield in a very simple
way the logarithmic Sobolev inequality on the same space. By an appropriate
integration by parts formula the proof also yields in the same way a
logarithmic Sobolev inequality for the path space equipped with a general
diffusion measure as long as the torsion of the corresponding Riemannian
connection satisfies Driver's total antisymmetry condition.
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