Regularity of solutions to anisotropic nonlocal equations
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Mathematische Zeitschrift
Regularity of solutions to anisotropic nonlocal equations Jamil Chaker1 Received: 27 April 2018 / Accepted: 18 December 2019 © The Author(s) 2020
Abstract We study harmonic functions associated to systems of stochastic differential equations of the j form d X ti = Ai1 (X t− )d Z t1 + · · · + Aid (X t− )d Z td , i ∈ {1, . . . , d}, where Z t are independent one-dimensional symmetric stable processes of order α j ∈ (0, 2), j ∈ {1, . . . , d}. In this article we prove Hölder regularity of bounded harmonic functions with respect to solutions to such systems. Keywords Jump processes · Harmonic functions · Hölder continuity · Support theorem · Anisotropy · Nonlocal Operators Mathematics Subject Classification Primary 60J75; Secondary 60H10 · 31B05 · 60G52
1 Introduction The consideration of stochastic processes with jumps and anisotropic behavior is natural and reasonable since such objects arise in several natural and financial models. In certain circumstances Lévy processes with jumps are more suitable to capture empirical facts that diffusion models do. See for instance [14] for examples of financial models with jumps. In the nineteen fifties, De Giorgi [15] and Nash [29] independently prove an a-priori Hölder estimate for weak solutions u to second order equations of the form div(A(x)∇u(x)) = 0 for uniformly elliptic and measurable coefficients A. In [28], Moser proves Hölder continuity of weak solutions and gives a proof of an elliptic Harnack inequality for weak solutions to this equation. This article provides a new technique of how to derive an a-priori Hölder estimate from the Harnack inequality. For a large class of local operators, the Hölder continuity can be derived from the Harnack inequality, see for instance [19]. For a comprehensive introduction into Harnack inequalities, we refer the reader e.g. to [20]. The corresponding case of operators in non-divergence form is treated in by Krylov and Safonov in [23]. The authors develop a technique for proving Hölder regularity and the
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Jamil Chaker [email protected] Universität Bielefeld, Fakultät für Mathematik, Postfach 100131, 33501 Bielefeld, Germany
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J. Chaker
Harnack inequality for harmonic functions corresponding to non-divergence form elliptic operators. They take a probabilistic point of view and make use of the martingale problem to prove regularity estimates for harmonic functions. The main tool is a support theorem, which gives information about the topological support for solutions to the martingale problem associated to the corresponding operator. This technique is also used in [6] to prove similar results for nonlocal operators of the form L f (x) = [ f (x + h) − f (x) − 1{|h|≤1} h · ∇ f (x)]a(x, h)dh (1.1) Rd \{0}
under suitable assumptions on the function a. In [4] Bass and Chen follow the same ideas to prove Hölder regularity for harmonic functions associated to solutions of systems of stochastic differential equations driven by Lévy processes with highly singular Lévy measures. In this wor
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