Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

Request Permissions   Purchase Content 


Gradient estimates of harmonic functions and transition densities for Lévy processes

Authors: Tadeusz Kulczycki and Michał Ryznar
Journal: Trans. Amer. Math. Soc. 368 (2016), 281-318
MSC (2010): Primary 31B05, 60J45; Secondary 60J50, 60J75
Published electronically: May 13, 2015
MathSciNet review: 3413864
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove gradient estimates for harmonic functions with respect to a $ d$-dimensional unimodal pure-jump Lévy process under some mild assumptions on the density of its Lévy measure. These assumptions allow for a construction of an unimodal Lévy process in $ \mathbb{R}^{d+2}$ with the same characteristic exponent as the original process. The relationship between the two processes provides a fruitful source of gradient estimates of transition densities. We also construct another process called a difference process which is very useful in the analysis of differential properties of harmonic functions. Our results extend the gradient estimates known for isotropic stable processes to a wide family of isotropic pure-jump processes, including a large class of subordinate Brownian motions.

References [Enhancements On Off] (What's this?)

  • [1] R. M. Blumenthal and R. K. Getoor, Markov processes and potential theory, Pure and Applied Mathematics, Vol. 29, Academic Press, New York-London, 1968. MR 0264757 (41 #9348)
  • [2] Krzysztof Bogdan, Tomasz Grzywny, and Michał Ryznar, Density and tails of unimodal convolution semigroups, J. Funct. Anal. 266 (2014), no. 6, 3543-3571. MR 3165234,
  • [3] K. Bogdan, T. Grzywny, M. Ryznar, Barriers, exit time and survival probability for unimodal Lévy processes, Probab. Theory Relat. Fields (2014), DOI 10.1007/s00440-014-0568-6.
  • [4] Krzysztof Bogdan and Tomasz Jakubowski, Estimates of heat kernel of fractional Laplacian perturbed by gradient operators, Comm. Math. Phys. 271 (2007), no. 1, 179-198. MR 2283957 (2007k:47076),
  • [5] K. Bogdan, T. Kulczycki, and Adam Nowak, Gradient estimates for harmonic and $ q$-harmonic functions of symmetric stable processes, Illinois J. Math. 46 (2002), no. 2, 541-556. MR 1936936 (2004b:60188)
  • [6] Krzysztof Bogdan, Takashi Kumagai, and Mateusz Kwaśnicki, Boundary Harnack inequality for Markov processes with jumps, Trans. Amer. Math. Soc. 367 (2015), no. 1, 477-517. MR 3271268,
  • [7] René Carmona, Path integrals for relativistic Schrödinger operators, Schrödinger operators (Sønderborg, 1988) Lecture Notes in Phys., vol. 345, Springer, Berlin, 1989, pp. 65-92. MR 1037317 (91g:60088),
  • [8] Zhen-Qing Chen, Panki Kim, and Renming Song, Dirichlet heat kernel estimates for fractional Laplacian with gradient perturbation, Ann. Probab. 40 (2012), no. 6, 2483-2538. MR 3050510,
  • [9] Zhen-Qing Chen, Panki Kim, and Renming Song, Dirichlet heat kernel estimates for rotationally symmetric Lévy processes, Proc. Lond. Math. Soc. (3) 109 (2014), no. 1, 90-120. MR 3237737,
  • [10] Zhen-Qing Chen, Panki Kim, and Renming Song, Heat kernel estimates for the Dirichlet fractional Laplacian, J. Eur. Math. Soc. (JEMS) 12 (2010), no. 5, 1307-1329. MR 2677618 (2012c:58058),
  • [11] Kai Lai Chung and Zhong Xin Zhao, From Brownian motion to Schrödinger's equation, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 312, Springer-Verlag, Berlin, 1995. MR 1329992 (96f:60140)
  • [12] M. Cranston and Z. Zhao, Some regularity results and eigenfunction estimates for the Schrödinger operator, Diffusion processes and related problems in analysis, Vol. I (Evanston, IL, 1989) Progr. Probab., vol. 22, Birkhäuser Boston, Boston, MA, 1990, pp. 139-147. MR 1110161 (92d:35080)
  • [13] Loukas Grafakos and Gerald Teschl, On Fourier transforms of radial functions and distributions, J. Fourier Anal. Appl. 19 (2013), no. 1, 167-179. MR 3019774,
  • [14] Tomasz Grzywny, On Harnack inequality and Hölder regularity for isotropic unimodal Lévy processes, Potential Anal. 41 (2014), no. 1, 1-29. MR 3225805,
  • [15] Nobuyuki Ikeda and Shinzo 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), 79-95. MR 0142153 (25 #5546)
  • [16] K. Kaleta, P. Sztonyk, Estimates of transition densities and their derivatives for jump Lévy processes, arXiv:1307.1302.
  • [17] Panki Kim and Ante Mimica, Harnack inequalities for subordinate Brownian motions, Electron. J. Probab. 17 (2012), no. 37, 23. MR 2928720
  • [18] Panki Kim, Renming Song, and Zoran Vondraček, Potential theory of subordinate Brownian motions revisited, Stochastic analysis and applications to finance, Interdiscip. Math. Sci., vol. 13, World Sci. Publ., Hackensack, NJ, 2012, pp. 243-290. MR 2986850,
  • [19] Panki Kim, Renming Song, and Zoran Vondraček, Two-sided Green function estimates for killed subordinate Brownian motions, Proc. Lond. Math. Soc. (3) 104 (2012), no. 5, 927-958. MR 2928332,
  • [20] Tadeusz Kulczycki, Gradient estimates of $ q$-harmonic functions of fractional Schrödinger operator, Potential Anal. 39 (2013), no. 1, 69-98. MR 3065315,
  • [21] Tadeusz Kulczycki and Bartłomiej Siudeja, Intrinsic ultracontractivity of the Feynman-Kac semigroup for relativistic stable processes, Trans. Amer. Math. Soc. 358 (2006), no. 11, 5025-5057. MR 2231884 (2007m:47098),
  • [22] Elliott H. Lieb and Robert Seiringer, The stability of matter in quantum mechanics, Cambridge University Press, Cambridge, 2010. MR 2583992 (2011j:81369)
  • [23] P. W. Millar, First passage distributions of processes with independent increments, Ann. Probability 3 (1975), 215-233. MR 0368177 (51 #4418)
  • [24] Ante Mimica, Harnack inequality and Hölder regularity estimates for a Lévy process with small jumps of high intensity, J. Theoret. Probab. 26 (2013), no. 2, 329-348. MR 3055806,
  • [25] Michał Ryznar, Estimates of Green function for relativistic $ \alpha $-stable process, Potential Anal. 17 (2002), no. 1, 1-23. MR 1906405 (2003f:60087),
  • [26] Ken-iti Sato, Lévy processes and infinitely divisible distributions, Cambridge Studies in Advanced Mathematics, vol. 68, Cambridge University Press, Cambridge, 1999. Translated from the 1990 Japanese original; Revised by the author. MR 1739520 (2003b:60064)
  • [27] René L. Schilling, Growth and Hölder conditions for the sample paths of Feller processes, Probab. Theory Related Fields 112 (1998), no. 4, 565-611. MR 1664705 (99m:60131),
  • [28] René L. Schilling, Paweł Sztonyk, and Jian Wang, Coupling property and gradient estimates of Lévy processes via the symbol, Bernoulli 18 (2012), no. 4, 1128-1149. MR 2995789,
  • [29] René L. Schilling, Renming Song, and Zoran Vondraček, Bernstein functions, Theory and applications, 2nd ed., de Gruyter Studies in Mathematics, vol. 37, Walter de Gruyter & Co., Berlin, 2012. MR 2978140
  • [30] Luis Silvestre, On the differentiability of the solution to an equation with drift and fractional diffusion, Indiana Univ. Math. J. 61 (2012), no. 2, 557-584. MR 3043588,
  • [31] Paweł Sztonyk, On harmonic measure for Lévy processes, Probab. Math. Statist. 20 (2000), no. 2, Acta Univ. Wratislav. No. 2256, 383-390. MR 1825650 (2002c:60126)
  • [32] Paweł Sztonyk, Regularity of harmonic functions for anisotropic fractional Laplacians, Math. Nachr. 283 (2010), no. 2, 289-311. MR 2604123 (2011e:47076),
  • [33] Toshiro Watanabe, The isoperimetric inequality for isotropic unimodal Lévy processes, Z. Wahrsch. Verw. Gebiete 63 (1983), no. 4, 487-499. MR 705619 (84m:60093),

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 31B05, 60J45, 60J50, 60J75

Retrieve articles in all journals with MSC (2010): 31B05, 60J45, 60J50, 60J75

Additional Information

Tadeusz Kulczycki
Affiliation: Institute of Mathematics and Computer Science, Wrocław University of Technology, Wyb. Wyspiańskiego 27, 50-370 Wrocław, Poland

Michał Ryznar
Affiliation: Institute of Mathematics and Computer Science, Wrocław University of Technology, Wyb. Wyspiańskiego 27, 50-370 Wrocław, Poland

Received by editor(s): August 1, 2013
Received by editor(s) in revised form: October 31, 2013
Published electronically: May 13, 2015
Additional Notes: This research was supported in part by NCN grant no. 2011/03/B/ST1/00423.
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society