Quasi-invariance and Gibbs structure of diffusion measures on infinite product groups

Abstract

We prove a quasi-invariance property for the distribution of solutions to a
stochastic differential equation on an infinite dimensional Lie group
${\mathbf G}$ constructed as the countable power of a compact Lie group $G$. In
the gradient case, we prove the Gibbs structure of the distribution and
construct the associated ''diffusion bridge'' measure on the loop space of ${\mathbf G}$.