Boundary Harnack inequality for Markov processes with jumps

Abstract

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

Martin Kernels for Markov Processes with Jumps

“…Note that by Proposition 2.1 in [12], under Assumptions 1 through 3, C exit (x 0 , r) is finite. We use the short-hand notation f ≈ cg for the two inequalities c −1 g ≤ f ≤ cg, where c > 0 is a positive constant.…”

“…In [12] the following four conditions are introduced. A detailed discussion of these assumptions is beyond the scope of the present article, we refer the reader to [12] for more information.…”

“…A detailed discussion of these assumptions is beyond the scope of the present article, we refer the reader to [12] for more information. Here we only state the conditions, without explaining in a formal way the notions of semi-polar and polar sets, processes in duality X t andX t , their generators A andÂ, densities ν(x, y) andν(x, y) (with respect to the measure m) of the Lévy kernels of X t andX t , as well as their Green functions G D (x, y) =Ĝ D (y, x).…”

“…The proof of our main result, Theorem 2, relies on the boundary Harnack inequality for Markov Mateusz Kwaśnicki mateusz.kwasnicki@pwr.edu.pl Tomasz Juszczyszyn tomasz.juszczyszyn@pwr.edu.pl 1 Faculty of Pure and Applied Mathematics, Wroclaw University of Science and Technology, Wroclaw, Poland processes with jumps, proved recently in [12], and the oscillation reduction argument, developed in [6] and [11]. As an application, we obtain Martin representation of harmonic functions in Theorem 3.…”

“…We conclude the introduction with a description of the structure of this article. The assumptions for the boundary Harnack inequality of [12] are briefly recalled in Section 2. We omit a detailed discussion of these conditions and refer the interested reader to the original paper.…”