Intrinsic ultracontractivity of nonsymmetric diffusions with measure-valued drifts and potentials

Abstract

Recently, in [Preprint (2006)], we extended the concept of intrinsic ultracontractivity to nonsymmetric semigroups. In this paper, we study the intrinsic ultracontractivity of nonsymmetric diffusions with measure-valued drifts and measure-valued potentials in bounded domains. Our process Y is a diffusion process whose generator can be formally written as L+μ⋅∇−ν with Dirichlet boundary conditions, where L is a uniformly elliptic second-order differential operator and μ=(μ¹, …, μ^d) is such that each component μⁱ, i=1, …, d, is a signed measure belonging to the Kato class K_d,1 and ν is a (nonnegative) measure belonging to the Kato class K_d,2. We show that scale-invariant parabolic and elliptic Harnack inequalities are valid for Y.

In this paper, we prove the parabolic boundary Harnack principle and the intrinsic ultracontractivity for the killed diffusion Y^D with measure-valued drift and potential when D is one of the following types of bounded domains: twisted Hölder domains of order α∈(1/3, 1], uniformly Hölder domains of order α∈(0, 2) and domains which can be locally represented as the region above the graph of a function. This extends the results in [J. Funct. Anal. 100 (1991) 181–206] and [Probab. Theory Related Fields 91 (1992) 405–443]. As a consequence of the intrinsic ultracontractivity, we get that the supremum of the expected conditional lifetimes of Y^D is finite.