HEAT KERNELS FOR REFLECTED DIFFUSIONS WITH JUMPS ON INNER UNIFORM DOMAINS

Zhen Qing Chen, Panki Kim, Takashi Kumagai, Jian Wang

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)

Abstract

In this paper, we study sharp two-sided heat kernel estimates for a large class of symmetric reflected diffusions with jumps on the closure of an inner uniform domain D in a length metric space. The length metric is the intrinsic metric of a strongly local Dirichlet form. When D is an inner uniform domain in the Euclidean space, a prototype for a special case of the processes under consideration is symmetric reflected diffusions with jumps on D, whose infinitesimal generators are non-local (pseudo-differential) operators L on D of the form Lu(x) = 1/2 d ∑ i,j=1 ∂/∂xi (aij(x) ∂u(x)/∂xj)) +lim ε0 ∫ {y∈D: ρD(y,x)>ε}(u(y)−u(x))J(x, y) dy satisfying “Neumann boundary condition”. Here, ρD(x, y) is the length metric on D, A(x) = (aij(x))1≤i,j≤d is a measurable d×d matrix-valued function on D that is uniformly elliptic and bounded, and J(x, y) := 1/Φ(ρD 1 (x, y)) ∫ [α1,α2] c(α, x, y)/ρD(x, y)d+α ν(dα) where ν is a finite measure on [α1, α2] ⊂ (0, 2), Φ is an increasing function on [0, ∞) with c1ec2rβ ≤ Φ(r) ≤ c3ec4 for some β ∈ [0, ∞], and c(α, x, y) is a jointly measurable function that is bounded between two positive constants and is symmetric in (x, y).

Original languageEnglish
Pages (from-to)6797-6841
Number of pages45
JournalTransactions of the American Mathematical Society
Volume375
Issue number10
DOIs
Publication statusPublished - 2022 Oct 1

Keywords

  • Reflected diffusions with jumps
  • inner uniform domain; heat kernel
  • symmetric Dirichlet form

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics

Fingerprint

Dive into the research topics of 'HEAT KERNELS FOR REFLECTED DIFFUSIONS WITH JUMPS ON INNER UNIFORM DOMAINS'. Together they form a unique fingerprint.

Cite this