The Dirichlet problem for stable-like operators and related probabilistic representations |
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| Authors: | Ari Arapostathis Anup Biswas Luis Caffarelli |
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| Institution: | 1. Department of Electrical and Computer Engineering, The University of Texas at Austin, Austin, TX, USAari@ece.utexas.edu;3. Department of Mathematics, Indian Institute of Science Education and Research, Pashan, Pune, Maharashtra, India;4. Department of Mathematics, The University of Texas at Austin, Austin, TX, USA |
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| 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. |
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| Keywords: | α-stable process Dirichlet problem exit time Harnack inequality invariant probability measure positive recurrence stochastic differential equations with jumps |
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