Heat kernel for non-local operators with variable order

Xin Chen, Zhen Qing Chen, Jian Wang*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

8 Citations (Scopus)

Abstract

Let α(x) be a measurable function taking values in [α12] for 0<α1⩽α2<2, and κ(x,z) be a positive measurable function that is symmetric in z and bounded between two positive constants. Under uniform Hölder continuous assumptions on α(x) and x↦κ(x,z), we obtain existence, upper and lower bounds, and regularity properties of the heat kernel associated with the following non-local operator of variable order [Formula presented]. In particular, we show that the operator L generates a conservative Feller process on Rd having strong Feller property, which is usually assumed a priori in the literature to study analytic properties of L via probabilistic approaches. Our near-diagonal estimates and lower bound estimates of the heat kernel depend on the local behavior of index function α(x). When α(x)≡α∈(0,2), our results recover some results by Chen and Kumagai (2003) and Chen and Zhang (2016).

Original languageEnglish
Pages (from-to)3574-3647
Number of pages74
JournalStochastic Processes and their Applications
Volume130
Issue number6
DOIs
Publication statusPublished - Jun 2020
Externally publishedYes

Keywords

  • Heat kernel
  • Levi's method
  • Non-local operator with variable order
  • Stable-like process

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