Wrapping Brownian motion and heat kernels on compact Lie groups

Wrapping Brownian motion and heat kernels II: symmetric spaces

In this paper we extend our previous results on wrapping Brownian motion and heat kernels onto compact Lie groups to various symmetric spaces, where a global generalisation of Rouvière'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.

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. This is achieved by considering Itô type stochastic differential equations and applying the Feynman-Kač theorem.

从Ben Saïd–Kobayashi–Ørsted (k,a)广义拉盖尔半群理论视角看热核与布朗运动的形变

Deformation of the heat kernel and the Brownian motion from the perspective of the Ben Saïd--Kobayashi--Ørsted $(k,a)$-generalized Laguerre semigroup theory

本研究基于Ben Saïd等人提出的(k,a)广义傅里叶分析，在k=0条件下构建ℝᴺ上的a形变热核与a形变布朗运动，探究其基本性质，并证明对应广义傅里叶积分核的多项式增长性。