Asymptotics for the heat kernel in multicone domains

A multicone domain Ω⊆RnΩ⊆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)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)limt→∞⁡t1+αp(t,x,y) in terms of the Martin boundary of Ω at infinity, where α>0α>0 depends on the geometry of Ω. We next derive an analogous result for tκ/2Px(T>t)tκ/2Px(T>t), with κ=1+α−n/2κ=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.