Asymptotics for the heat kernel in multicone domains

Abstract

A multicone domain Omega subset of 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 ->infinity)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)P(x) (T > t), with kappa = 1 +alpha-n/2, where T is the exit time from Omega. Lastly, we deduce the renormalized Yaglom limit for the process conditioned on survival.