Absolute Continuity of Heat Kernel Measure with Pinned Wiener Measure on Loop Groups

Abstract

Let $t > 0, K$ be a connected compact Lie group equipped with an $Ad_K$- invariant inner product on the Lie Algebra of $K$. Associated to this data are two measures $\mu^0_t$ and $\nu^0_t$ on $\mathscr{L}(K)$ – the space of continuous loops based at $e \in K$. The measure $\mu^0_t$ is pinned Wiener measure with “variance $t$” while the measure $\nu^0_t$ is a “heat kernel measure” on $\mathscr{L}(K)$. The measure $\mu^0_t$ is constructed using a $K$-valued Brownian motion while the measure $\nu^0_t$ is constructed using a $\mathscr{L}(K)$-valued Brownian motion. In this paper we show that $\nu^0_t$ is absolutely continuous with respect to $\mu^0_t$ and the Radon-Nikodym derivative $d\nu^0_t /d\mu^0_t$ is bounded.