Long Time Asymptotics of Heat Kernels and Brownian Winding Numbers on Manifolds with Boundary

TL;DR

This paper analyzes the long-term behavior of heat kernels and Brownian winding numbers on manifolds with boundary, providing precise asymptotics and a Gaussian limit theorem for winding fluctuations.

Contribution

It derives exact long-time asymptotics of heat kernels on manifolds with boundary and establishes a central limit theorem for Brownian winding numbers in this setting.

Findings

Exact asymptotic formulas for heat kernels on manifolds with boundary.

A Gaussian central limit theorem for Brownian winding numbers.

Explicit covariance matrix for winding fluctuations.

Abstract

Let M be a compact Riemannian manifold with smooth boundary. We obtain the exact long time asymptotic behaviour of the heat kernel on abelian coverings of M with mixed Dirichlet and Neumann boundary conditions. As an application, we study the long time behaviour of the abelianized winding of reflected Brownian motions in M. In particular, we prove a Gaussian type central limit theorem showing that when rescaled appropriately, the fluctuations of the abelianized winding are normally distributed with an explicit covariance matrix.