The log-Sobolev inequality on loop space over a compact Riemannian manifold (Q1268078)

The authors 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 depends only on the Ricci curvature of the manifold and the Hessian of the heat kernel, and is \(L^p\)-integrable for all \(p\geq 1\). The log-Sobolev inequality is obtained as a consequence of a martingale representation theorem for the differentiable functions on the loop space which is a variation of the Clark-Ocone-Haussman formula.