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

Zhen-Qing Chen and Eryan Hu

Stochastic Processes and their Applications, 2015, vol. 125, issue 7, 2603-2642

Abstract: For α∈(0,2) and M>0, we consider a family of nonlocal operators {Δ+aαΔα/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.

Keywords: Heat kernel; Transition density; Feller semigroup; Perturbation; Positivity; Lévy system; Kato class (search for similar items in EconPapers)
Date: 2015
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Citations: View citations in EconPapers (2)

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DOI: 10.1016/j.spa.2015.02.016

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