Asymptotics for the heat kernel in multicone domains

Original · EN

A multi cone 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 ₜ→∞ 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 form Ω. Lastly, we deduce the renormalized Yaglom limit for the process conditioned on survival.