Estimates on Green functions and Schrödinger-type equations for non-symmetric diffusions with measure-valued drifts

Panki Kim, Renming Song

Research output: Contribution to journalArticlepeer-review

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 ṡ ∇ + 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.

Original languageEnglish (US)
Pages (from-to)57-80
Number of pages24
JournalJournal of Mathematical Analysis and Applications
Volume332
Issue number1
DOIs
StatePublished - Aug 1 2007

Keywords

  • 3G theorem
  • Boundary Harnack principle
  • Brownian motion
  • Diffusion
  • Diffusion process
  • Green function
  • Harmonic function
  • Heat kernel
  • Kato class
  • Lipschitz domain
  • Measure-valued drift
  • Non-symmetric diffusion
  • Schrödinger operator
  • Transition density

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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