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September 2008 Intrinsic ultracontractivity of nonsymmetric diffusions with measure-valued drifts and potentials
Panki Kim, Renming Song
Ann. Probab. 36(5): 1904-1945 (September 2008). DOI: 10.1214/07-AOP381


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.


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Panki Kim. Renming Song. "Intrinsic ultracontractivity of nonsymmetric diffusions with measure-valued drifts and potentials." Ann. Probab. 36 (5) 1904 - 1945, September 2008.


Published: September 2008
First available in Project Euclid: 11 September 2008

zbMATH: 1175.47039
MathSciNet: MR2440927
Digital Object Identifier: 10.1214/07-AOP381

Primary: 47D07 , 60J25
Secondary: 60J45

Keywords: Diffusions , Dual processes , Harnack inequality , intrinsic ultracontractivity , nonsymmetric diffusions , nonsymmetric semigroups , parabolic boundary Harnack principle , parabolic Harnack inequality , semigroups

Rights: Copyright © 2008 Institute of Mathematical Statistics


Vol.36 • No. 5 • September 2008
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