Green function for gradient perturbation of unimodal Lévy processes

• Autorzy: Grzywny T. , Jakubowski T. , Żurek G.

Identyfikatory

DOI

10.19195/0208-4147.37.1.5

• Języki publikacji: EN

Abstrakty

EN

We prove that the Green function of a generator of isotropic unimodal Lévy processes with the weak lower scaling order greater than one and the Green function of its gradient perturbations are comparable for bounded smooth open sets if the drift function is from an appropriate Kato class.

Słowa kluczowe

EN

unimodal Lévy process; heat kernel; smooth domain; Green function; gradient perturbation

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2017
• Tom: Vol. 37, Fasc. 1
• Strony: 119--143
• Opis fizyczny: Bibliogr. 31 poz.

Twórcy

autor

Grzywny T.

• Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland

autor

Jakubowski T.

• Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland

autor

Żurek G.

• Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland

Bibliografia

• [1] R. M. Blumenthal and R. K. Getoor, Markov Processes and Potential Theory, Pure and Applied Mathematics, Vol. 29, Academic Press, New York 1968.
• [2] K. Bogdan, Y. Butko, and K. Szczypkowski, Majorization, 4G Theorem and Schrödinger perturbations, J. Evol. Equ. 16 (2016), pp. 241-260.
• [3] K. Bogdan, T. Grzywny, and M. Ryznar, Density and tails of unimodal convolution semigroups, J. Funct. Anal. 266 (6) (2014), pp. 3543-3571.
• [4] K. Bogdan, T. Grzywny, and M. Ryznar, Dirichlet heat kernel for unimodal Lévy processes, Stochastic Process. Appl. 124 (11) (2014), pp. 3612-3650.
• [5] K. Bogdan and T. Jakubowski, Estimates of heat kernel of fractional Laplacian perturbed by gradient operators, Comm. Math. Phys. 271 (1) (2007), pp. 179-198.
• [6] K. Bogdan and T. Jakubowski, Estimates of the Green function for the fractional Laplacian perturbed by gradient, Potential Anal. 36 (3) (2012), pp. 455-481.
• [7] K. Bogdan and T. Komorowski, Principal eigenvalue of the fractional Laplacian with a large incompressible drift, NoDEA Nonlinear Differential Equations Appl. 21 (4) (2014), pp. 541-566.
• [8] L. A. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math. (2) 171 (3) (2010), pp. 1903-1930.
• [9] Z.-Q. Chen, On notions of harmonicity, Proc. Amer. Math. Soc. 137 (10) (2009), pp. 3497-3510.
• [10] Z.-Q. Chen, P. Kim, and R. Song, Dirichlet heat kernel estimates for fractional Laplacian with gradient perturbation, Ann. Probab. 40 (6) (2012), pp. 2483-2538.
• [11] Z.-Q. Chen, P. Kim, and R. Song, Dirichlet heat kernel estimates for rotationally symmetric Lévy processes, Proc. Lond. Math. Soc. (3) 109 (1) (2014), pp. 90-120.
• [12] Z.-Q. Chen, Y.-X. Ren, and T. Yang, Boundary Harnack principle and gradient estimates for fractional Laplacian perturbed by non-local operators, Potential Anal. 45 (3) (2016), pp. 509-537.
• [13] K. L. Chung and Z. X. Zhao, From Brownian Motion to Schrödinger’s Equation, Grundlehren Math. Wiss., Vol. 312, Springer, Berlin 1995.
• [14] P. Graczyk, T. Jakubowski, and T. Luks, Martin representation and Relative Fatou Theorem for fractional Laplacian with a gradient perturbation, Positivity 17 (4) (2013), pp. 1043-1070.
• [15] T. Grzywny, On Harnack inequality and Hölder regularity for isotropic unimodal Lévy processes, Potential Anal. 41 (1) (2014), pp. 1-29.
• [16] T. Grzywny and K. Szczypkowski, Estimates of heat kernels of non-symmetric jump processes, preprint 2016.
• [17] N. Ikeda and S. Watanabe, On some relations between the harmonic measure and the Lévy measure for a certain class of Markov processes, J. Math. Kyoto Univ. 2 (1962), pp. 79-95.
• [18] T. Jakubowski, On Harnack inequality for α-stable Ornstein-Uhlenbeck processes, Math. Z. 258 (3) (2008), pp. 609-628.
• [19] T. Jakubowski, Fractional Laplacian with singular drift, Studia Math. 207 (3) (2011), pp. 257-273.
• [20] T. Jakubowski and K. Szczypkowski, Estimates of gradient perturbation series, J. Math. Anal. Appl. 389 (1) (2012), pp. 452-460.
• [21] P. Kim, R. Song, and Z. Vondraček, Martin boundary for some symmetric Lévy processes, in: Festschrift Masatoshi Fukushima, Z.-Q. Chen, N. Jacob, M. Takeda, and T. Uemura (Eds.), World Scientific, 2015, pp. 307-342.
• [22] A. Kiselev, F. Nazarov, and A. Volberg, Global well-posedness for the critical 2D dissipative quasi-geostrophic equation, Invent. Math. 167 (3) (2007), pp. 445-453.
• [23] T. Kulczycki and M. Ryznar, Gradient estimates of harmonic functions and transition densities for Lévy processes, Trans. Amer. Math. Soc. 368 (1) (2016), pp. 281-318.
• [24] Y. Maekawa and H. Miura, Upper bounds for fundamental solutions to non-local diffusion equations with divergence free drift, J. Funct. Anal. 264 (10) (2013), pp. 2245-2268.
• [25] A. Petrosyan and C. A. Pop, Optimal regularity of solutions to the obstacle problem for the fractional Laplacian with drift, J. Funct. Anal. 268 (2) (2015), pp. 417-472.
• [26] N. I. Portenko, Some perturbations of drift-type for symmetric stable processes, Random Oper. Stoch. Equ. 2 (3) (1994), pp. 211-224.
• [27] W. E. Pruitt, The growth of random walks and Lévy processes, Ann. Probab. 9 (6) (1981), pp. 948-956.
• [28] L. Silvestre, Hölder estimates for advection fractional-diffusion equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 11 (4) (2012), pp. 843-855.
• [29] L. Silvestre, V. Vicol, and A. Zlatoš, On the loss of continuity for super-critical drift-diffusion equations, Arch. Ration. Mech. Anal. 207 (3) (2013), pp. 845-877.
• [30] P. Sztonyk, On harmonic measure for Lévy processes, Probab. Math. Statist. 20 (2) (2000), pp. 383-390.
• [31] T. Watanabe, The isoperimetric inequality for isotropic unimodal Lévy processes, Z. Wahrsch. Verw. Gebiete 63 (4) (1983), pp. 487-499.

Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).