Asymptotics for the heat kernel in multicone domains

A multi cone domain $Ω\subseteq \mathbb{R}^n$ 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 $Ω$, using both probabilistic and analytical techniques. We find that the decay is polynomial and we characterize $\lim_{t\to\infty} t^{1+α}p(t,x,y)$ in terms of the Martin boundary of $Ω$ at infinity, where $α>0$ depends on the geometry of $Ω$. We next derive an analogous result for $t^{κ/2}\mathbb{P}_x(T >t)$, with $κ= 1+α- n/2$, where $T$ is the exit time form $Ω$. Lastly, we deduce the renormalized Yaglom limit for the process conditioned on survival.

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