HEAT KERNELS FOR REFLECTED DIFFUSIONS WITH JUMPS ON INNER UNIFORM DOMAINS

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 _ε↓_{0 ∫ {y∈D: ρD(y,x)>ε}}(u(y)−u(x))J(x, y) dy satisfying “Neumann boundary condition”. Here, ρ_D(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 c1e^c2rβ ≤ Φ(r) ≤ c3e^c4rβ 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).