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 $L+\mu \cdot \nabla -\nu$ with Dirichlet boundary conditions, where $L$ is a uniformly elliptic second-order differential operator and $\mu =({\mu }^1,...,{\mu }^d)$ is such that each component ${\mu }^i$, $i=1,...,d$, is a signed measure belonging to the Kato class ${\mathbf{K}}_{d,1}$ and $\nu$ is a (nonnegative) measure belonging to the Kato class ${\mathbf{K}}_{d,2}$. 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 $\alpha \in (1/3,1]$, uniformly Hölder domains of order $\alpha \in (0,2)$ 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.

Published in at http://dx.doi.org/10.1214/07-AOP381 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)