Feeling boundary by Brownian motion in a ball

TL;DR

This paper analyzes the short-time behavior of the Laplace Dirichlet heat kernel in a ball, providing precise boundary asymptotics and rates of convergence, extending classical principles of boundary non-influence.

Contribution

It offers detailed asymptotic formulas and convergence rates for the heat kernel in a ball, enhancing understanding of boundary effects in heat propagation.

Findings

Precise short-time asymptotics for the heat kernel

Explicit boundary behavior description

Rates of convergence established

Abstract

We provide short-time asymptotics with rates of convergence for the Laplace Dirichlet heat kernel in a ball. The boundary behaviour is precisely described. Presented results may be considered as a complement or a generalization of the famous "principle of not feeling the boundary" in case of a ball.