Asymptotics for the heat kernel in multicone domains

TL;DR

This paper investigates the long-time decay behavior of the heat kernel in multicone domains, revealing polynomial decay rates and characterizing limits using the domain's geometric and boundary properties.

Contribution

It provides a detailed analysis of heat kernel decay in multicone domains, linking decay rates to the Martin boundary and geometric features, and derives related probabilistic limits.

Findings

Heat kernel decays polynomially in multicone domains.

Decay rate characterized by Martin boundary at infinity.

Derived Yaglom limit for conditioned process.

Abstract

A multi cone domain $\Omega \subseteq \mathbb{R}^n$ is an open, connected set that resembles a finite collection of cones far away from the origin. We study the rate of decay in time of the heat kernel $p(t,x,y)$ of a Brownian motion killed upon exiting $\Omega$, using both probabilistic and analytical techniques. We find that the decay is polynomial and we characterize $\lim_{t\to\infty} t^{1+\alpha}p(t,x,y)$ in terms of the Martin boundary of $\Omega$ at infinity, where $\alpha>0$ depends on the geometry of $\Omega$. We next derive an analogous result for $t^{\kappa/2}\mathbb{P}_x(T >t)$, with $\kappa = 1+\alpha - n/2$, where $T$ is the exit time form $\Omega$. Lastly, we deduce the renormalized Yaglom limit for the process conditioned on survival.