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 = (μ 1 ,... μ d ) is such that each component μ i , i = 1,..., d, is a signed measure belonging to the Kato class K d , and v 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 Holder domains of order a e (1/3, 1], uniformly Holder domains of order a ∈ (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.