On dual processes of non-symmetric diffusions with measure-valued drifts

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• 作者：Panki Kim ; Renming Song
• 关键词：Diffusion ; Diffusion process ; Non-symmetric diffusion ; Dual process ; Brownian motion ; Kato class ; Measure-valued drift ; Transition density ; Green function ; Boundary Harnack principle ; Martin boundary ; Harmonic measure ; Schrö ; dinger operator
• 刊名：Stochastic Processes and their Applications
• 出版年：2008
• 出版时间：May 2008
• 年：2008
• 卷：118
• 期：5
• 页码：790-817
• 全文大小：492 K

文摘

For μ=(μ¹,…,μ^d) with each μⁱ being a signed measure on belonging to the Kato class , a diffusion with drift μ is a diffusion process in whose generator can be formally written as L+μ where L is a uniformly elliptic differential operator. When each μⁱ is given by for some function Uⁱ, a diffusion with drift μ is a diffusion in with generator L+U. In [P. Kim, R. Song, Two-sided estimates on the density of Brownian motion with singular drift, Illinois J. Math. 50 (2006) 635–688; P. Kim, R. Song, Boundary Harnack principle for Brownian motions with measure-valued drifts in bounded Lipschitz domains, Math. Ann., 339 (1) (2007) 135–174], we have already studied properties of diffusions with measure-valued drifts in bounded domains. In this paper we first show that the killed diffusion process with measure-valued drift in any bounded domain has a dual process with respect to a certain reference measure. We then discuss the potential theory of the dual process and Schrödinger-type operators of a diffusion with measure-valued drift. More precisely, we prove that (1) for any bounded domain, a scale invariant Harnack inequality is true for the dual process; (2) if the domain is bounded C^1,1, the boundary Harnack principle for the dual process is valid and the (minimal) Martin boundary for the dual process can be identified with the Euclidean boundary; and (3) the harmonic measure for the dual process is locally comparable to that of the h-conditioned Brownian motion with h being an eigenfunction corresponding to the largest Dirichlet eigenvalue in the domain.