The Log-Sobolev Inequality on Loop Space over a Compact Riemannian Manifold

Abstract

We obtain a log-Sobolev inequality with a neat and explicit potential for the gradient on a based loop space over a compact Riemannian manifold. The potential term relies only on the curvature of the manifold and the Hessian of the heat kernel, and isLp-integrable for allp⩾1. The log-Sobolev inequality is derived by a martingale representation theorem for the differentiable functions on loop space, which is a variation of the Clark–Ocone–Haussmann formula.