Asymptotics for the heat kernel in multicone domains

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 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 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.