Boundary harnack inequality for Markov processes with jumps

Transactions of the American Mathematical Society

Volumn 367, Issue 1, 2015, Pages 477-517

  Bogdan, Krzysztof ^a,b   Kumagai, Takashi ^c   Kwaśnicki, Mateusz ^a,b  

^a INSTITUTE OF MATHEMATICS   (Poland)

^b WROCLAW UNIVERSITY OF TECHNOLOGY   (Poland)

^c KYOTO UNIVERSITY   (Japan)

Author keywords

Boundary harnack inequality; Jump Markov process

Indexed keywords

EID: 84924787452     PISSN: 00029947     EISSN: 10886850     Source Type: Journal    
DOI: 10.1090/s0002-9947-2014-06127-8     Document Type: Article

References (83)

1. Hiroaki Aikawa, Boundary Harnack principle and Martin boundary for a uniform domain, J. Math. Soc. Japan 53 (2001), no. 1, 119-145, DOI 10.2969/jmsj/05310119. MR1800526 (2001m:31007)
2. Hiroaki Aikawa, Equivalence between the boundary Harnack principle and the Carleson estimate, Math. Scand. 103 (2008), no. 1, 61-76. MR2464701 (2009j:31008)
3. Hiroaki Aikawa, Boundary Harnack principle and the quasihyperbolic boundary condition, Sobolev spaces in mathematics. II, Int. Math. Ser. (N. Y.), vol. 9, Springer, New York, 2009, pp. 19-30, DOI 10.1007/978-0-387-85650-63. MR2484620 (2010b:31009)
4. Alano Ancona, Principe de Harnack à la frontière et théorème de Fatou pour un opérateur elliptique dans un domaine lipschitzien (French, with English summary), Ann. Inst. Fourier (Grenoble) 28 (1978), no. 4, 169-213, x. MR513885 (80d:31006)
5. Rodrigo Bañuelos and Krzysztof Bogdan, Symmetric stable processes in cones, Potential Anal. 21 (2004), no. 3, 263-288, DOI 10.1023/B:POTA.0000033333.72236.dc. MR2075671 (2005h:60238)
6. Guy Barles, Emmanuel Chasseigne, and Cyril Imbert, Hölder continuity of solutions of second-order non-linear elliptic integro-differential equations, J. Eur. Math. Soc. (JEMS) 13 (2011), no. 1, 1-26, DOI 10.4171/JEMS/242. MR2735074 (2011j:45008)
7. Martin T. Barlow and Richard F. Bass, Transition densities for Brownian motion on the Sierpiński carpet, Probab. Theory Related Fields 91 (1992), no. 3-4, 307-330, DOI 10.1007/BF01192060. MR1151799 (93k:60203)
8. Martin T. Barlow and Richard F. Bass, Brownian motion and harmonic analysis on Sierpiński carpets, Canad. J. Math. 51 (1999), no. 4, 673-744, DOI 10.4153/CJM-1999-031-4. MR1701339 (2000i:60083)
9. Martin T. Barlow, Richard F. Bass, Zhen-Qing Chen, and Moritz Kassmann, Non-local Dirichlet forms and symmetric jump processes, Trans. Amer. Math. Soc. 361 (2009), no. 4, 1963-1999, DOI 10.1090/S0002-9947-08-04544-3. MR2465826 (2010e:60163)
10. Martin T. Barlow, Alexander Grigoryan, and Takashi Kumagai, Heat kernel upper bounds for jump processes and the first exit time, J. Reine Angew. Math. 626 (2009), 135-157, DOI 10.1515/CRELLE.2009.005. MR2492992 (2009m:58077)
11. Martin T. Barlow and Edwin A. Perkins, Brownian motion on the Sierpiński gasket, Probab. Theory Related Fields 79 (1988), no. 4, 543-623, DOI 10.1007/BF00318785. MR966175 (89g:60241)
12. Richard F. Bass, Adding and subtracting jumps from Markov processes, Trans. Amer. Math. Soc. 255 (1979), 363-376, DOI 10.2307/1998181. MR542886 (81b:60070)
13. Richard F. Bass and Krzysztof Burdzy, A probabilistic proof of the boundary Harnack principle, Seminar on Stochastic Processes, 1989 (San Diego, CA, 1989), Progr. Probab., vol. 18, Birkhäuser Boston, Boston, MA, 1990, pp. 1-16. MR1042338 (92c:60106)
14. Richard F. Bass and Krzysztof Burdzy, A boundary Harnack principle in twisted Hölder domains, Ann. of Math. (2) 134 (1991), no. 2, 253-276, DOI 10.2307/2944347. MR1127476 (92m:31006)
15. Richard F. Bass and Krzysztof Burdzy, The boundary Harnack principle for nondivergence form elliptic operators, J. London Math. Soc. (2) 50 (1994), no. 1, 157-169, DOI 10.1112/jlms/50.1.157. MR1277760 (95f:35058)
16. Richard F. Bass and David A. Levin, Harnack inequalities for jump processes, Potential Anal. 17 (2002), no. 4, 375-388, DOI 10.1023/A:1016378210944. MR1918242 (2003e:60194)
17. Albert Benveniste and Jean Jacod, Systèmes de Lévy des processus de Markov (French), Invent. Math. 21 (1973), 183-198. MR0343375 (49 #8117)
18. Robert M. Blumenthal and Ronald K. Getoor, Markov processes and potential theory, Pure and Applied Mathematics, Vol. 29, Academic Press, New York, 1968. MR0264757 (41 #9348)
19. Krzysztof Bogdan, The boundary Harnack principle for the fractional Laplacian, Studia Math. 123 (1997), no. 1, 43-80. MR1438304 (98g:31005)
20. Krzysztof Bogdan, Sharp estimates for the Green function in Lipschitz domains, J. Math. Anal. Appl. 243 (2000), no. 2, 326-337, DOI 10.1006/jmaa.1999.6673. MR1741527 (2001b:31007)
21. Krzysztof Bogdan, Krzysztof Burdzy, and Zhen-Qing Chen, Censored stable processes, Probab. Theory Related Fields 127 (2003), no. 1, 89-152, DOI 10.1007/s00440-003-0275-1. MR2006232 (2004g:60068)
22. Krzysztof Bogdan and Tomasz Byczkowski, Potential theory for the α-stable Schrödinger operator on bounded Lipschitz domains, Studia Math. 133 (1999), no. 1, 53-92. MR1671973 (99m:31010)
23. Krzysztof Bogdan, Tomasz Byczkowski, Tadeusz Kulczycki, Michal Ryznar, RenMing Song, and Zoran Vondracek, Potential analysis of stable processes and its extensions, Lecture Notes in Mathematics, vol. 1980, Springer-Verlag, Berlin, 2009. Edited by Piotr Graczyk and Andrzej Stos. MR2569321 (2011i:60140)
24. Krzysztof Bogdan, Tomasz Grzywny, and Michał Ryznar, Heat kernel estimates for the fractional Laplacian with Dirichlet conditions, Ann. Probab. 38 (2010), no. 5, 1901-1923, DOI 10.1214/10-AOP532. MR2722789 (2011f:60151)
25. Krzysztof Bogdan and Tomasz Jakubowski, Estimates of the Green function for the fractional Laplacian perturbed by gradient, Potential Analysis 36 (2012), no. 3, 455-481. MR2892584
26. Krzysztof Bogdan, Tadeusz Kulczycki, and Mateusz Kwaśnicki, Estimates and structure of α-harmonic functions, Probab. Theory Related Fields 140 (2008), no. 3-4, 345-381, DOI 10.1007/s00440-007-0067-0. MR2365478 (2008k:31019)
27. Krzysztof Bogdan, Andrzej Stós, and Paweł Sztonyk, Harnack inequality for stable processes on d-sets, Studia Math. 158 (2003), no. 2, 163-198, DOI 10.4064/sm158-2-5. MR2013738 (2005i:60092)
28. Krzysztof Bogdan and Paweł Sztonyk, Estimates of the potential kernel and Harnack’s inequality for the anisotropic fractional Laplacian, Studia Math. 181 (2007), no. 2, 101-123, DOI 10.4064/sm181-2-1. MR2320691 (2008k:31015)
29. Zhen-Qing Chen and Panki Kim, Green function estimate for censored stable processes, Probab. Theory Related Fields 124 (2002), no. 4, 595-610, DOI 10.1007/s00440-002-0226-2. MR1942325 (2003m:60212)
30. Zhen-Qing Chen, Panki Kim, and Takashi Kumagai, Weighted Poincaré inequality and heat kernel estimates for finite range jump processes, Math. Ann. 342 (2008), no. 4, 833-883, DOI 10.1007/s00208-008-0258-8. MR2443765 (2010b:60230)
31. 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, DOI 10.4171/JEMS/231. MR2677618 (2012c:58058)
32. Zhen-Qing Chen, Panki Kim, RenMing Song, and Zoran Vondracek, Boundary Harnack principle for Δ + Δα/2, Trans. Amer. Math. Soc. 364 (2012), no. 8, 4169-4205, DOI 10.1090/S0002-9947-2012-05542-5. MR2912450
33. Zhen-Qing Chen and Takashi Kumagai, Heat kernel estimates for stable-like processes on dsets, Stochastic Process. Appl. 108 (2003), no. 1, 27-62, DOI 10.1016/S0304-4149(03)00105-4. MR2008600 (2005d:60135)
34. Zhen-Qing Chen and Takashi Kumagai, A priori Hölder estimate, parabolic Harnack principle and heat kernel estimates for diffusions with jumps, Rev. Mat. Iberoam. 26 (2010), no. 2, 551-589, DOI 10.4171/RMI/609. MR2677007 (2011h:47093)
35. Zhen-Qing Chen and Kazuhiro Kuwae, On subhamonicity for symmetric Markov processes, J. Math. Soc. Japan. 64 (2012), no. 4, 1181-1209. MR2998921
36. Zhen-Qing Chen and RenMing Song, Estimates on Green functions and Poisson kernels for symmetric stable processes, Math. Ann. 312 (1998), no. 3, 465-501, DOI 10.1007/s002080050232. MR1654824 (2000b:60179)
37. Kai Lai Chung and John B. Walsh, Markov processes, Brownian motion, and time symmetry, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 249, Springer, New York, 2005. MR2152573 (2006j:60003)
38. Björn E. J. Dahlberg, Estimates of harmonic measure, Arch. Rational Mech. Anal. 65 (1977), no. 3, 275-288. MR0466593 (57 #6470)
39. Claude Dellacherie and Paul-André Meyer, Probabilities and potential. C, North-Holland Mathematics Studies, vol. 151, North-Holland Publishing Co., Amsterdam, 1988. Potential theory for discrete and continuous semigroups; Translated from the French by J. Norris. MR939365 (89b:60002)
40. Eugene B. Dynkin, Markov processes. Vols. I, II, Translated with the authorization and assistance of the author by J. Fabius, V. Greenberg, A. Maitra, G. Majone. Die Grundlehren der Mathematischen Wissenschaften, Bände 121, vol. 122, Academic Press Inc., Publishers, New York, 1965. MR0193671 (33 #1887)
41. Pat J. Fitzsimmons, Ben M. Hambly, and Takashi Kumagai, Transition density estimates for Brownian motion on affine nested fractals, Comm. Math. Phys. 165 (1994), no. 3, 595-620. MR1301625 (95j:60122)
42. David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR737190 (86c:35035)
43. Alexander Grigoryan, Heat kernel and analysis on manifolds, AMS/IP Studies in Advanced Mathematics, vol. 47, American Mathematical Society, Providence, RI, 2009. MR2569498 (2011e:58041)
44. Qing-Yang Guan, Boundary Harnack inequalities for regional fractional Laplacian. ArXiv e-prints, May 2007.
45. Qing-Yang Guan and Zhi-Ming Ma, Reflected symmetric α-stable processes and regional fractional Laplacian, Probab. Theory Related Fields 134 (2006), no. 4, 649-694, DOI 10.1007/s00440-005-0438-3. MR2214908 (2007a:60034)
46. Pavel Gyrya and Laurent Saloff-Coste, Neumann and Dirichlet heat kernels in inner uniform domains (English, with English and French summaries), Astérisque 336 (2011), viii+144. MR2807275 (2012m:35117)
47. Wolfhard Hansen, Uniform boundary Harnack principle and generalized triangle property, J. Funct. Anal. 226 (2005), no. 2, 452-484, DOI 10.1016/j.jfa.2004.12.010. MR2160104 (2006i:35004)
48. Wolfhard Hansen, Three views on potential theory. Lecture notes for a course given at Charles University (Prague) in 2008. Available at http://www.karlin.mff.cuni.cz/~hansen/lecture/course-07012009.pdf, 2009.
49. Walter Hoh, Pseudo differential operators generating Markov processes, Habilitation, Universit ät Bielefeld, 1998. Available at http://www.mathematik.uni-bielefeld.de/~hoh/pdo_mp.ps.
50. Walter Hoh, A symbolic calculus for pseudo-differential operators generating Feller semigroups, Osaka J. Math. 35 (1998), no. 4, 789-820. MR1659620 (99k:47102)
51. 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. MR0142153 (25 #5546)
52. Tomasz Jakubowski, The estimates for the Green function in Lipschitz domains for the symmetric stable processes, Probab. Math. Statist. 22 (2002), no. 2, Acta Univ. Wratislav. No. 2470, 419-441. MR1991120 (2004g:60069) Licensed to Elsevier BV. Prepared on Thu Nov 20 23:31:27 EST 2014 for download from
53. Tomasz Jakubowski, Fractional Laplacian with singular drift, Studia Math. 207 (2011), no. 3, 257-273, DOI 10.4064/sm207-3-3. MR2875353 (2012k:60226)
54. David S. Jerison and Carlos E. Kenig, Boundary value problems on Lipschitz domains, Studies in partial differential equations, MAA Stud. Math., vol. 23, Math. Assoc. America, Washington, DC, 1982, pp. 1-68. MR716504 (85f:35057)
55. Kamil Kaleta and Mateusz Kwaśnicki, Boundary Harnack inequality for α-harmonic functions on the Sierpiński triangle, Probab. Math. Statist. 30 (2010), no. 2, 353-368. MR2792590 (2012f:60271)
56. Moritz Kassmann, A priori estimates for integro-differential operators with measurable kernels, Calc. Var. Partial Differential Equations 34 (2009), no. 1, 1-21, DOI 10.1007/s00526-008-0173-6. MR2448308 (2010b:35474)
57. John T. Kemper, A boundary Harnack principle for Lipschitz domains and the principle of positive singularities, Comm. Pure Appl. Math. 25 (1972), 247-255. MR0293114 (45 #2193)
58. John T. Kemper, A correction to: “A boundary Harnack principle for Lipschitz domains and the principle of positive singularities” (Comm. Pure Appl. Math. 25 (1972), 247-255). Comm. Pure Appl. Math. 29 (1976), no. 5, 555. MR0422644 (54:10630)
59. Jun Kigami, Analysis on fractals, Cambridge Tracts in Mathematics, vol. 143, Cambridge University Press, Cambridge, 2001. MR1840042 (2002c:28015)
60. Panki Kim and Ante Mimica, Harnack inequalities for subordinate Brownian motions, Electron. J. Probab. 17 (2012), no. 37, 23. MR2928720
61. Panki Kim and Ren Ming Song, Potential theory of truncated stable processes, Math. Z. 256 (2007), no. 1, 139-173, DOI 10.1007/s00209-006-0063-6. MR2282263 (2007k:60245)
62. Panki Kim and RenMing Song, Boundary behavior of harmonic functions for truncated stable processes, J. Theoret. Probab. 21 (2008), no. 2, 287-321, DOI 10.1007/s10959-008-0145-y. MR2391246 (2010f:60230)
63. Panki Kim, RenMing Song, and Zoran Vondraček, Boundary Harnack principle for subordinate Brownian motions, Stochastic Process. Appl. 119 (2009), no. 5, 1601-1631, DOI 10.1016/j.spa.2008.08.003. MR2513121 (2010f:60231)
64. Panki Kim, RenMing Song, and Zoran Vondracek, 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. MR2986850
65. Panki Kim, RenMing Song, and Zoran Vondraček, Uniform boundary Harnack principle for rotationally symmetric Lévy processes in general open sets, Sci. China Math. 55 (2012), no. 11, 2317-2333, DOI 10.1007/s11425-012-4516-6. MR2994122
66. 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. MR2928332
67. Tadeusz Kulczycki, Properties of Green function of symmetric stable processes, Probab. Math. Statist. 17 (1997), no. 2, Acta Univ. Wratislav. No. 2029, 339-364. MR1490808 (98m:60119)
68. Takashi Kumagai, Estimates of transition densities for Brownian motion on nested fractals, Probab. Theory Related Fields 96 (1993), no. 2, 205-224, DOI 10.1007/BF01192133. MR1227032 (94e:60068)
69. Takashi Kumagai, Some remarks for stable-like jump processes on fractals, Fractals in Graz 2001, Trends Math., Birkhäuser, Basel, 2003, pp. 185-196. MR2091704 (2006a:60138)
70. P. Joseph McKenna and Wolfgang Reichel, A priori bounds for semilinear equations and a new class of critical exponents for Lipschitz domains, J. Funct. Anal. 244 (2007), no. 1, 220-246, DOI 10.1016/j.jfa.2006.11.018. MR2294482 (2007k:35158)
71. Luke G. Rogers, Robert S. Strichartz, and Alexander Teplyaev, Smooth bumps, a Borel theorem and partitions of smooth functions on P.C.F. fractals, Trans. Amer. Math. Soc. 361 (2009), no. 4, 1765-1790, DOI 10.1090/S0002-9947-08-04772-7. MR2465816 (2010f:28020)
72. 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. MR1739520 (2003b:60064)
73. René L. Schilling, RenMing Song, and Zoran Vondraček, Bernstein functions, de Gruyter Studies in Mathematics, vol. 37, Walter de Gruyter & Co., Berlin, 2010. Theory and applications. MR2598208 (2011d:60060) 74] Michael Sharpe, General theory of Markov processes, Pure and Applied Mathematics, vol. 133, Academic Press Inc., Boston, MA, 1988. MR958914 (89m:60169)
74. RenMing Song and Jang-Mei Wu, Boundary Harnack principle for symmetric stable processes, J. Funct. Anal. 168 (1999), no. 2, 403-427, DOI 10.1006/jfan.1999.3470. MR1719233 (2001b:60092)
75. Andrzej Stós, Symmetric α-stable processes on d-sets, Bull. Polish Acad. Sci. Math. 48 (2000), no. 3, 237-245. MR1779007 (2002f:60152)
76. Andrzej Stós, Boundary Harnack principle for fractional powers of Laplacian on the Sierpiński carpet (English, with English and French summaries), Bull. Sci. Math. 130 (2006), no. 7, 580-594, DOI 10.1016/j.bulsci.2006.01.001. MR2261965 (2008h:60320)
77. Robert S. Strichartz and Michael Usher, Splines on fractals, Math. Proc. Cambridge Philos. Soc. 129 (2000), no. 2, 331-360, DOI 10.1017/S0305004100004424. MR1765920 (2001c:28016)
78. Paweł Sztonyk, Approximation of stable-dominated semigroups, Potential Anal. 33 (2010), no. 3, 211-226, DOI 10.1007/s11118-009-9165-1. MR2718253 (2011i:60158)
79. Shinzo Watanabe, On discontinuous additive functionals and Lévy measures of a Markov process, Japan. J. Math. 34 (1964), 53-70. MR0185675 (32 #3137)
80. Takesi Watanabe, On the equivalence of excessive functions and superharmonic functions in the theory of Markov processes. I, Proc. Japan Acad. 38 (1962), 397-401. MR0150831 (27 #818a)
81. Toshiro Watanabe, Asymptotic estimates of multi-dimensional stable densities and their applications, Trans. Amer. Math. Soc. 359 (2007), no. 6, 2851-2879 (electronic), DOI 10.1090/S0002-9947-07-04152-9. MR2286060 (2008e:60143)
82. Jang Mei G. Wu, Comparisons of kernel functions, boundary Harnack principle and relative Fatou theorem on Lipschitz domains (English, with French summary), Ann. Inst. Fourier (Grenoble) 28 (1978), no. 4, 147-167, vi. MR513884 (80g:31005)
83. Zhong Xin Zhao, Green function for Schrödinger operator and conditioned Feynman-Kac gauge, J. Math. Anal. Appl. 116 (1986), no. 2, 309-334, DOI 10.1016/S0022-247X(86)80001-4. MR842803 (88f:60142)