Abstract
Recently, in [Preprint (2006)], we extended the concept of intrinsic ultra-contractivity 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 Holder domains of order α ∈ (1/3,1], uniformly Holder 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.
Original language | English (US) |
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Pages (from-to) | 1904-1945 |
Number of pages | 42 |
Journal | Annals of Probability |
Volume | 36 |
Issue number | 5 |
DOIs | |
State | Published - Sep 2008 |
Keywords
- Diffusions
- Dual processes
- Harnack inequality
- Intrinsic ultracontractivity
- Non-symmetric semigroups
- Nonsymmetric diffusions
- Parabolic boundary harnack principle
- Parabolic harnack inequality
- Semigroups
ASJC Scopus subject areas
- Statistics and Probability
- Statistics, Probability and Uncertainty