Estimates for Dirichlet Heat Kernels, Intrinsic Ultracontractivity and Expected Exit Time on Lipschitz Domains

Abstract

We prove pointwise estimates for the heat kernel of a secondorder elliptic operator with Dirichlet boundary conditions on a bounded Lipschitz domain in $\mathbb{R}^n,\, n\geq 1$. Applications to obtain estimates for intrinsic ultracontractivity of the heat semigroup and expected exit time of a Brownian motion are given.