Wrapping Brownian motion and heat kernels I: compact Lie groups

TL;DR

This paper explores how to derive heat kernels on compact Lie groups by wrapping Brownian motion from their Lie algebra using stochastic differential equations and the wrapping map.

Contribution

It introduces a method to obtain heat kernels on compact Lie groups from Euclidean space via the wrapping map, linking stochastic processes and harmonic analysis.

Findings

Derived heat kernels on compact Lie groups using wrapping map.

Connected Brownian motion on Lie algebra to heat kernels on Lie groups.

Applied stochastic differential equations and Feynman-Kac theorem in the analysis.

Abstract

An important object of study in harmonic analysis is the heat equation. On a Euclidean space, the fundamental solution of the associated semigroup is known as the heat kernel, which is also the law of Brownian motion. Similar statements also hold in the case of a Lie group. By using the wrapping map of Dooley and Wildberger, we show how to wrap a Brownian motion to a compact Lie group from its Lie algebra (viewed as a Euclidean space) and find the heat kernel. This is achieved by considering It\^o type stochastic differential equations and applying the Feynman-Ka\v{c} theorem.