Heat Kernel Analysis on Infinite-Dimensional Heisenberg Groups

TL;DR

This paper develops heat kernel analysis on infinite-dimensional Heisenberg groups, establishing bounds, invariance, and inequalities for heat kernel measures, with implications for stochastic analysis on these groups.

Contribution

It introduces a new class of infinite-dimensional Heisenberg-like groups and derives key geometric and probabilistic properties of their heat kernels.

Findings

Heat kernel measures have Gaussian-like upper bounds.

Cameron-Martin quasi-invariance holds for these measures.

Logarithmic Sobolev inequalities are established.

Abstract

We introduce a class of non-commutative Heisenberg like infinite dimensional Lie groups based on an abstract Wiener space. The Ricci curvature tensor for these groups is computed and shown to be bounded. Brownian motion and the corresponding heat kernel measures, $\{\nu_t\}_{t>0},$ are also studied. We show that these heat kernel measures admit: 1) Gaussian like upper bounds, 2) Cameron-Martin type quasi-invariance results, 3) good $L^p$ -- bounds on the corresponding Radon-Nykodim derivatives, 4) integration by parts formulas, and 5) logarithmic Sobolev inequalities. The last three results heavily rely on the boundedness of the Ricci tensor.