No abstract

In this paper, we investigate symmetric jump-type processes on a class of metric measure spaces with jumping intensities comparable to radially symmetric functions on the spaces. The class of metric measure spaces includes the Alfors d-regular sets, which is a class of fractal sets that contains geometrically self-similar sets. A typical example of our jump-type processes is the symmetric jump process with jumping intensitywhere ν is a probability measure on [α 1 , α 2 ] ⊂ (0, 2), c(α, x, y) is a jointly measurable function that is symmetric in (x, y) and is bounded between two positive constants, and c 0 (x, y) is a jointly measurable function that is symmetric in (x, y) and is bounded between γ 1 and γ 2 , where either γ 2 ≥ γ 1 > 0 or γ 1 = γ 2 = 0. This example contains mixed symmetric stable processes on R n as well as mixed relativistic symmetric stable Dedicated to Professor Masatoshi Fukushima on the occasion of his 70th birthday.

Let (X, d, µ) be a metric measure space with a local regular Dirichlet form. We give necessary and sufficient conditions for a parabolic Harnack inequality with global spacetime scaling exponent β ≥ 2 to hold. We show that this parabolic Harnack inequality is stable under rough isometries. As a consequence, once such a Harnack inequality is established on a metric measure space, then it holds for any uniformly elliptic operator in divergence form on a manifold naturally defined from the graph approximation of the space.

We prove a boundary Harnack inequality for jump-type Markov processes on metric measure state spaces, under comparability estimates of the jump kernel and Urysohn-type property of the domain of the generator of the process. The result holds for positive harmonic functions in arbitrary open sets. It applies, e.g., to many subordinate Brownian motions, L\'evy processes with and without continuous part, stable-like and censored stable processes, jump processes on fractals, and rather general Schr\"odinger, drift and jump perturbations of such processes.Comment: 37 pages, 1 figure, minor editorial changes, paper accepted in Transactions of AM

Abstract. In this paper, we study sharp heat kernel estimates for a large class of symmetric jump-type processes in R d for all t > 0. A prototype of the processes under consideration are symmetric jump processes on R d with jumping intensitywhere ν is a probability measure on [α 1 , α 2 ] ⊂ (0, 2), Φ is an increasing function on [0, ∞) with c 1 e c 2 r β ≤ Φ(r) ≤ c 3 e c 4 r β with 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). They include, in particular, mixed relativistic symmetric stable processes on R d with different masses. We also establish the parabolic Harnack principle.