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

• 成果类型: Article
• 署名作者: Kim, Panki; Song, Renming
• 署名单位: Seoul National University (SNU); University of Illinois System; University of Illinois Urbana-Champaign
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/07-AOP381
• 发表日期: 2008
• 页码: 1904-1945
• 关键词: brownian-motion; SCHRODINGER-OPERATORS; harnack inequality; singular drift; heat kernel; sets

摘要

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 + mu . del - v with Dirichlet boundary conditions, where L is a uniformly elliptic second-order differential operator and mu = (mu(l).....mu(d)) is such that each component mu(i), i = l,....d. is a signed measure belonging to the Kato class K-d,K-l and v is a (nonnegative) measure belonging to the Kato class K-d,K-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 alpha is an element of (1/3, 1], uniformly Holder domains of order alpha is an element of (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 intrinsec ultracontractivity, we get that the supremum of the expected conditional lifetimes of Y-D is finite.