## 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 = (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, 1} and ν 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 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 Y^{D} 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