Brownian motion and heat kernels on compact lie groups and symmetric spaces.

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 ildberger, 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 Ito type stochastic differential equations and applying the Feynman-Kaˇc theorem. We also consider wrapping Brownian motion to various symmetric spaces, where a global generalisation of Rouviere's formula and the e-function are considered. Additionally, we extend some of our results to complex Lie groups, and certain non-compact symmetric spaces.