Discussions on heat kernel measure and pinned wiener measure

Abstract

The log-Sobolev inequalities on loop spaces over compact Riemannian manifolds have been received much attention (see [1-6]). In particular, for loop group over a compact type Lie group, by replacing pinne~i Wiener measure with a heat kernel measure, Driver and Lohrenz [2] obtained a log-Sobolev inequality without added potential. Let G be a d-dimensional compact type Lie group with unit element e and harr measure dx. Denote by g the Lie algebra of G with an Ad(G) invariant inner product (-, ")9" Set