Quasi-invariance and Gibbs structure of diffusion measures on infinite product groups (Q2711776)

The authors consider stochastic differential equations on the infinite dimensional Lie group \(G\) constructed as a countable power of a compact Lie group. It is shown that the distributions of solutions are quasi-invariant with respect to a natural dense subgroup. The Gibbs structure of the distribution is established in the case where the drift coefficient is given by the logarithmic derivative of a Gibbs measure. Diffusion ``bridge measures'' are constructed on the loop space of \(G\), and identified with Gibbs measures of the Euclidean type on the infinite product of loop groups.