Heat kernels for reflected diffusions with jumps on inner uniform domains

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 are symmetric reflected diffusions with jumps on D, whose infinitesimal generators are non-local (pseudo-differential) operators L on D of the form(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, andwhere ν is a finite measure on [α1, α2] ⊂ (0, 2), Φ is an increasing function on [0, ∞) with c1e c 2 r β ≤ Φ(r) ≤ c3e c 4 r β 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).