Regularity of fundamental solutions for Lévy-type operators

• Autorzy: Szczypkowski Karol

Identyfikatory

DOI

10.4064/ba230424-12-9

• Języki publikacji: EN

Abstrakty

EN

For a class of non-symmetric non-local Lévy-type operators Lκ, which include those of the form Lκf(x) := Rd(f(x + z) − f(x) − 1|z|<1⟨z, ∇f(x)⟩)κ(x, z)J(z) dz, e prove regularity of the fundamental solution pκ to the equation ∂t = Lκ.

Słowa kluczowe

EN

Lévy-type operator; Hölder continuity; gradient estimates; fundamental solution; heat kernel; non-symmetric operator; non-local operator; non-symmetric; Markov process; Levi’s parametrix method

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2023
• Tom: Vol. 71, no 2
• Strony: 169--191
• Opis fizyczny: Bibliogr. 23 poz.

Twórcy

autor

Szczypkowski Karol

• Wydział Matematyki, Politechnika Wrocławska, 50-370 Wrocław, Poland

• https://orcid.org/0000-0001-9986-0788

Bibliografia

• [1] K. Bogdan, P. Sztonyk, and V. Knopova, Heat kernel of anisotropic nonlocal operators, Doc. Math. 25 (2020), 1-54.
• [2] B. Böttcher, A parametrix construction for the fundamental solution of the evolution equation associated with a pseudo-differential operator generating a Markov process, Math. Nachr. 278 (2005), 1235-1241.
• [3] Z.-Q. Chen and X. Zhang, Heat kernels and analyticity of non-symmetric jump diffusion semigroups, Probab. Theory Related Fields 165 (2016), 267-312.
• [4] K. Du and X. Zhang, Optimal gradient estimates of heat kernels of stable-like operators, Proc. Amer. Math. Soc. 147 (2019), 3559-3565.
• [5] S. D. Eidelman, S. D. Ivasyshen, and A. N. Kochubei, Analytic Methods in the Theory of Differential and Pseudo-Differential Equations of Parabolic Type, Operator Theory Adv. Appl. 152, Birkhäuser, Basel, 2004.
• [6] C. L. Epstein and C. A. Pop, Regularity for the supercritical fractional Laplacian with drift, J. Geom. Anal. 26 (2016), 1231-1268.
• [7] T. Grzywny and K. Szczypkowski, Heat kernels of non-symmetric Lévy-type operators, J. Differential Equations 267 (2019), 6004-6064.
• [8] T. Grzywny and K. Szczypkowski, Estimates of heat kernels of non-symmetric Lévy processes, Forum Math. 33 (2021), 1207-1236.
• [9] P. Jin, Heat kernel estimates for non-symmetric stable-like processes, arXiv:1709.02836 (2017).
• [10] P. Kim, R. Song, and Z. Vondraček, Heat kernels of non-symmetric jump processes: beyond the stable case, Potential Anal. 49 (2018), 37-90.
• [11] V. Knopova and A. Kulik, Parametrix construction for certain Lévy-type processes, Random Oper. Stoch. Equ. 23 (2015), 111-136.
• [12] V. Knopova and A. Kulik, Intrinsic compound kernel estimates for the transition probability density of Lévy-type processes and their applications, Probab. Math. Statist. 37 (2017), 53-100.
• [13] V. Knopova and A. Kulik, Parametrix construction of the transition probability density of the solution to an SDE driven by α-stable
• [14] A. Kohatsu-Higa and L. Li, Regularity of the density of a stable-like driven SDE with Hölder continuous coefficients, Stoch. Anal. Appl. 34 (2016), 979-1024.
• [15] F. Kühn, Transition probabilities of Lévy-type processes: parametrix construction, Math. Nachr. 292 (2019), 358-376.
• [16] T. Kulczycki and M. Ryznar, Gradient estimates of harmonic functions and transition densities for Lévy processes, Trans. Amer. Math. Soc. 368 (2016), 281-318.
• [17] T. Kulczycki and M. Ryznar, Transition density estimates for diagonal systems of SDEs driven by cylindrical α-stable processes, ALEA Lat. Amer. J. Probab. Math. Statist. 15 (2018), 1335-1375.
• [18] E. E. Levi, Sulle equazioni lineari totalmente ellittiche alle derivate parziali, Rend. Circ. Mat. Palermo 24 (1907), 275-317.
• [19] W. Liu, R. Song, and L. Xie, Gradient estimates for the fundamental solution of Lévy type operator, Adv. Nonlinear Anal. 9 (2020), 1453-1462.
• [20] W. Rudin, Real and Complex Analysis, 3rd ed., McGraw-Hill, New York, 1987.
• [21] R. L. Schilling, P. Sztonyk, and J. Wang, Coupling property and gradient estimates of Lévy processes via the symbol, Bernoulli 18 (2012), 1128-1149.
• [22] K. Szczypkowski, Fundamental solution for super-critical non-symmetric Lévy-type operators, Adv. Differential Equations, to appear; arXiv:1807.04257.
• [23] L. Xie and X. Zhang, Heat kernel estimates for critical fractional diffusion operators, Studia Math. 224 (2014), 221-263.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2024)