Heat kernel analysis on semi-infinite Lie groups

TL;DR

This paper investigates heat kernel measures and Brownian motion on semi-infinite Lie groups, establishing quasi-invariance, $L^p$ estimates, and a logarithmic Sobolev inequality in an infinite-dimensional setting.

Contribution

It introduces new quasi-invariance results, $L^p$ bounds, and Sobolev inequalities for heat kernels on semi-infinite Lie groups, expanding understanding of infinite-dimensional analysis.

Findings

Proved a Cameron-Martin type quasi-invariance theorem.

Derived $L^p$ norm estimates for Radon-Nikodym derivatives.

Established a logarithmic Sobolev inequality in this context.

Abstract

This paper studies Brownian motion and heat kernel measure on a class of infinite dimensional Lie groups. We prove a Cameron-Martin type quasi-invariance theorem for the heat kernel measure and give estimates on the $L^p$ norms of the Radon-Nikodym derivatives. We also prove that a logarithmic Sobolev inequality holds in this setting.