Lifetime and Heat Kernel Estimates in Non-Smooth Domains

Abstract

Let D be a domain in R n, n ≥ 2, and let B t be Brownian motion in D with lifetime τD. The transition probabilities for this motion are given by the Dirichlet heat kernel P t D(x, y) for \(\frac{1}{2}\Delta\) in D. If h is a positive harmonic function in D the Doob h-process, the Brownian motion conditioned by h, is determined by the following transition functions: $$P_t^h(x,y) = \frac{1}{h(x)}P_t^D(x,y)h(y)$$ .