The Annals of Probability

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

Panki Kim and Renming Song

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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, …, μd) is such that each component μi, i=1, …, d, is a signed measure belonging to the Kato class Kd,1 and ν is a (nonnegative) measure belonging to the Kato class Kd,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 YD 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 YD is finite.

Article information

Ann. Probab., Volume 36, Number 5 (2008), 1904-1945.

First available in Project Euclid: 11 September 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47D07: Markov semigroups and applications to diffusion processes {For Markov processes, see 60Jxx} 60J25: Continuous-time Markov processes on general state spaces
Secondary: 60J45: Probabilistic potential theory [See also 31Cxx, 31D05]

Diffusions nonsymmetric diffusions dual processes semigroups nonsymmetric semigroups Harnack inequality parabolic Harnack inequality parabolic boundary Harnack principle intrinsic ultracontractivity


Kim, Panki; Song, Renming. Intrinsic ultracontractivity of nonsymmetric diffusions with measure-valued drifts and potentials. Ann. Probab. 36 (2008), no. 5, 1904--1945. doi:10.1214/07-AOP381.

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