Asymptotics for the heat kernel in multicone domains

Abstract

A multicone domain Ω ⊆ Rn 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 limt→∞ t1+αp(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 tκ/2Px(T >t), with κ = 1 + α − n/2, where T is the exit time from Ω. Lastly, we deduce the renormalized Yaglom limit for the process conditioned on survival.