Martingale Representation and a Simple Proof of Logarithmic Sobolev Inequalities on Path Spaces

Mireille Capitaine,Michel Ledoux,Elton Hsu

doi:10.1214/ecp.v2-986

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.

Highlights

Let ω =t≥0 be a standard Brownian motion starting from the origin with values in IRn and denote by W0(IRn) the path space of continuous functions from [0, 1] to IRn starting from the origin
Thanks to the linear structure of W0(IRn), the proof of the logarithmic Sobolev inequality (1) may be reduced to the case of finite dimensional Gaussian measures, for which rather elementary semigroup arguments may be used. Such semigroup arguments can be formulated in terms of stochastic calculus on Brownian paths, which may be shown to work in the infinite dimensional setting as well
In the last section we show how the previous stochastic calculus argument may be applied to yield the isoperimetric inequality on path spaces in [B-L]

Summary

Introduction

The aim of this note is to observe that, together with this representation formula, the preceding simple proof of the logarithmic Sobolev inequality for Brownian motion in IRn yields in exactly the same way the logarithmic Sobolev inequality for the law of Brownian motion on a complete Riemannian manifold with bounded Ricci curvature. In the last section we show how the previous stochastic calculus argument may be applied to yield the isoperimetric inequality on path spaces in [B-L]

Logarithmic Sobolev Inequality for Brownian Motion on a Manifold

Diffusion Processes and Connections with Torsion

Isoperimetric Inequalities on Path Spaces