The Dirichlet problem for stable-like operators and related probabilistic representations. Issue 9 (1st September 2016)

• Record Type: Journal Article
• Authors: Arapostathis, Ari; Biswas, Anup; Caffarelli, Luis
• Is Part Of: Communications in partial differential equations. Volume 41:Issue 9(2016)
• Page Start: 1472
• Page End: 1511
• Publication Date: 2016-09-01
• Subjects: α-stable process -- Dirichlet problem -- exit time -- Harnack inequality -- invariant probability measure -- positive recurrence -- stochastic differential equations with jumps; Primary 60J45, 60J75, 58F11 -- Secondary 60H30; Differential equations, Partial -- Periodicals; 515.353
• DOI: 10.1080/03605302.2016.1207084
• Languages: English
• ISSNs: 0360-5302

Abstract

ABSTRACT: We study stochastic differential equations with jumps with no diffusion part, governed by a large class of stable-like operators, which may contain a drift term. For this class of operators, we establish the regularity of solutions to the Dirichlet problem up to the boundary as well as the usual stochastic characterization of these solutions. We also establish key connections between the recurrence properties of the jump process and the associated nonlocal partial differential operator. Provided that the process is positive (Harris) recurrent, we also show that the mean hitting time of a ball is a viscosity solution of an exterior Dirichlet problem.