Heat kernel estimates for Δ+Δα/2 under gradient perturbation

Zhen Qing Chen, Eryan Hu*

*Corresponding author for this work

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Abstract

Abstract For α∈(0,2) and M>0, we consider a family of nonlocal operators {Δ+Δα/2,a∈(0,M]} on Rd under Kato class gradient perturbation. We establish the existence and uniqueness of their fundamental solutions, and derive their sharp two-sided estimates. The estimates give explicit dependence on a and recover the sharp estimates for Brownian motion with drift as a→0. Each fundamental solution determines a conservative Feller process X. We characterize X as the unique solution of the corresponding martingale problem as well as a Lévy process with singular drift.

Original languageEnglish
Article number2761
Pages (from-to)2603-2642
Number of pages40
JournalStochastic Processes and their Applications
Volume125
Issue number7
DOIs
Publication statusPublished - Jul 2015
Externally publishedYes

Keywords

  • Feller semigroup
  • Heat kernel
  • Kato class
  • Lévy system
  • Perturbation
  • Positivity
  • Transition density

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Chen, Z. Q., & Hu, E. (2015). Heat kernel estimates for Δ+Δα/2 under gradient perturbation. Stochastic Processes and their Applications, 125(7), 2603-2642. Article 2761. https://doi.org/10.1016/j.spa.2015.02.016