Estimates on Green functions and Schrödinger-type equations for non-symmetric diffusions with measure-valued drifts |
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Authors: | Panki Kim Renming Song |
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Affiliation: | a Department of Mathematics, Seoul National University, Seoul 151-742, Republic of Korea b Department of Mathematics, University of Illinois, Urbana, IL 61801, USA |
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Abstract: | In this paper, we establish sharp two-sided estimates for the Green functions of non-symmetric diffusions with measure-valued drifts in bounded Lipschitz domains. As consequences of these estimates, we get a 3G type theorem and a conditional gauge theorem for these diffusions in bounded Lipschitz domains.Informally the Schrödinger-type operators we consider are of the form L+μ⋅∇+ν where L is a uniformly elliptic second order differential operator, μ is a vector-valued signed measure belonging to Kd,1 and ν is a signed measure belonging to Kd,2. In this paper, we establish two-sided estimates for the heat kernels of Schrödinger-type operators in bounded C1,1-domains and a scale invariant boundary Harnack principle for the positive harmonic functions with respect to Schrödinger-type operators in bounded Lipschitz domains. |
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Keywords: | Brownian motion Diffusion Diffusion process Non-symmetric diffusion Kato class Measure-valued drift Transition density Green function Lipschitz domain 3G theorem Schrö dinger operator Heat kernel Boundary Harnack principle Harmonic function |
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