Wrapping Brownian Motion and Heat Kernels. I: Compact Lie Groups

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.