Martin kernels for Markov processes with jumps. (English) Zbl 1380.31008
The paper discusses the existence of boundary limits of ratios of positive harmonic functions for a large class of Markov processes featuring jumps and irregular domains, in the context of general metric measure spaces. As a by-product, the authors prove the uniqueness of the Martin kernel at each boundary point and thus, the Martin boundary can be identified with the topological boundary. Several examples are included: strictly stable Lévy processes with positive continuous density of the Lévy measure; stable-like processes in the whole space or in domains; stable-like subordinate diffusions in metric measure spaces.
Reviewer: Marius Ghergu (Dublin)
MSC:
| 31C35 | Martin boundary theory |
| 60J45 | Probabilistic potential theory |
| 60J50 | Boundary theory for Markov processes |
| 60J75 | Jump processes (MSC2010) |
References:
| [1] | Aikawa, H.: Boundary Harnack principle and Martin boundary for a uniform domain. J. Math. Soc. Japan 53(1), 119-145 (2001) · Zbl 0976.31002 · doi:10.2969/jmsj/05310119 |
| [2] | Aikawa, H., Boundary Harnack Principle and the Quasihyperbolic Boundary Condition, 19-30 (2009), New York · Zbl 1166.31002 · doi:10.1007/978-0-387-85650-6_3 |
| [3] | Aikawa, H.: Extended Harnack inequalities with exceptional sets and a boundary Harnack principle. J. Anal. Math. 124, 83-116 (2014) · Zbl 1314.31008 · doi:10.1007/s11854-014-0028-3 |
| [4] | Aikawa, H.: Intrinsic ultracontractivity via capacitary width. Rev. Mat. Iberoamericana 31(3), 1041-1106 (2015) · Zbl 1331.31005 · doi:10.4171/RMI/863 |
| [5] | Ancona, A.: Principe de Harnack à la frontière et théorème de Fatou pour un opérateur elliptique dans un domaine lipschitzien. Ann. Inst. Fourier 28(4), 169-213 (1978) · Zbl 0377.31001 · doi:10.5802/aif.720 |
| [6] | Bogdan, K.: The boundary Harnack principle for the fractional Laplacian. Studia Math. 123(1), 43-80 (1997) · Zbl 0870.31009 · doi:10.4064/sm-123-1-43-80 |
| [7] | Bogdan, K.: Representation of α-harmonic functions in Lipschitz domains. Hiroshima Math. J. 29(2), 227-243 (1999) · Zbl 0936.31008 |
| [8] | Bogdan, K., Burdzy, K., Chen, Z.-Q.: Censored stable processes. Probab. Theory Relat. Fields 127(1), 89-152 (2003) · Zbl 1032.60047 · doi:10.1007/s00440-003-0275-1 |
| [9] | Bogdan, K., Grzywny, T., Ryznar, M.: Density and tails of unimodal convolution semigroups. J. Funct. Anal. 266(6), 3543-3571 (2014) · Zbl 1290.60002 · doi:10.1016/j.jfa.2014.01.007 |
| [10] | Bogdan, K., Grzywny, T., Ryznar, M.: Barriers, exit time and survival probability for unimodal Lévy processes. Probab. Theory Relat. Fields 162(1-2), 155-198 (2015) · Zbl 1317.31016 · doi:10.1007/s00440-014-0568-6 |
| [11] | Bogdan, K., Kulczycki, T., Kwaśnicki, M.: Estimates and structure of α-harmonic functions. Probab. Theory Relat. Fields 140(3-4), 345-381 (2008) · Zbl 1146.31004 |
| [12] | Bogdan, K., Kumagai, T., Kwaśnicki, M.: Boundary Harnack inequality for Markov processes with jumps. Trans. Amer. Math. Soc. 367, 477-517 (2015) · Zbl 1309.60080 · doi:10.1090/S0002-9947-2014-06127-8 |
| [13] | Chen, Z. -Q., Kim, P., Song, R., Vondraček, Z.: Boundary Harnack principle for Δ+Δα/2. Trans. Amer. Math. Soc. 364, 4169-4205 (2012) · Zbl 1271.31006 · doi:10.1090/S0002-9947-2012-05542-5 |
| [14] | Chen, Z. -Q., Song, R.: Martin Boundary and Integral Representation for Harmonic Functions of Symmetric Stable Processes. J. Funct. Anal. 159(1), 267-294 (1998) · Zbl 0954.60003 · doi:10.1006/jfan.1998.3304 |
| [15] | Chung, KL, Doubly-Feller Process with Multiplicative Functional, 63-78 (1986), Boston · Zbl 0603.60066 · doi:10.1007/978-1-4684-6748-2_4 |
| [16] | Chung, K. L., Walsh, J. B.: Markov Processes, Brownian Motion, and Time Symmetry Grundlehren Der Mathematischen Wissenschaften, vol. 249. Springer, New York (2005) · Zbl 0954.60003 |
| [17] | Dahlberg, B. E. J.: Estimates of harmonic measure. Arch. Rational Mech. Anal. 65(3), 275-288 (1977) · Zbl 0406.28009 · doi:10.1007/BF00280445 |
| [18] | Grzywny, T.: On Harnack inequality and Hölder regularity for isotropic unimodal Lévy processes. Potential Anal. 41(1), 1-29 (2014) · Zbl 1302.60109 · doi:10.1007/s11118-013-9360-y |
| [19] | Grzywny, T., Kwaśnicki, M.: Potential kernels, probabilities of hitting a ball, harmonic functions and the boundary Harnack inequality for unimodal Lévy processes. arXiv:1611.10304 (2016) · Zbl 1386.60265 |
| [20] | Grzywny, T., Ryznar, M.: Hitting times of points and intervals for symmetric Lévy processes. To appear in Potential Analysis. doi:10.1007/s11118-016-9600-z(2016) · Zbl 1391.60106 |
| [21] | Guan, Q.: Boundary Harnack inequalities for regional fractional Laplacian. arXiv:0705.1614v3 (2007) · Zbl 1317.31016 |
| [22] | Kim, P., Song, R., Vondraček, Z.: Boundary Harnack principle for subordinate Brownian motions. Stoch. Process. Appl. 119(5), 1601-1631 (2009) · Zbl 1166.60046 · doi:10.1016/j.spa.2008.08.003 |
| [23] | Kim, P., Song, R., Vondraček, Z.: Uniform boundary Harnack principle for rotationally symmetric Lévy processes in general open sets. Sci. China Math. 55(11), 2317-2333 (2012) · Zbl 1262.60077 · doi:10.1007/s11425-012-4516-6 |
| [24] | Kim, P., Song, R., Vondraček, Z.: Potential theory of subordinate Brownian motions with Gaussian components. Stoch. Process. Appl. 123(3), 764-795 (2013) · Zbl 1266.31007 · doi:10.1016/j.spa.2012.11.007 |
| [25] | Kim, P., Song, R., Vondraček, Z.: Global uniform boundary Harnack principle with explicit decay rate and its application. Stoch. Process. Appl. 124(1), 235-267 (2014) · Zbl 1296.60198 · doi:10.1016/j.spa.2013.07.007 |
| [26] | Kim, P., Song, R., Vondraček, Z.: Boundary Harnack principle and Martin boundary at infinity for subordinate Brownian motions. Potential Anal. 41, 407-441 (2014) · Zbl 1302.60110 · doi:10.1007/s11118-013-9375-4 |
| [27] | Kim, P., Song, R., Vondraček, Z.: Martin Boundary for Some Symmetric Lévy Processes Festschrift Masatoshi Fukushima, In Honor of Masatoshi Fukushima’s Sanju, Interdisciplinary Mathematical, Sciences, vol. 17. World Scientific (2015) · Zbl 0976.31002 |
| [28] | Kim, P., Song, R., Vondraček, Z.: Scale invariant boundary Harnack principle at infinity for Feller processes. To appear in Potential Analysis. arXiv:1510.04569v2 (2015) · Zbl 1381.60110 |
| [29] | Kim, P., Song, R., Vondraček, Z.: Accessibility, Martin boundary and minimal thinness for Feller processes in metric measure spaces. To appear in Revista Matemática Iberoamericana. arXiv:1510.04571v3 (2015) · Zbl 1395.60094 |
| [30] | Kim, P., Song, R., Vondraček, Z.: Martin boundary of unbounded sets for purely discontinuous Feller processes. Forum Math. 28, 1067-1086 (2016). doi:10.1515/forum-2015-0233 (2015) · Zbl 1351.60106 |
| [31] | Lierl, J., Saloff-Coste, L.: Scale-invariant boundary Harnack principle in inner uniform domains. Osaka J. Math. 51, 619-657 (2014) · Zbl 1301.31008 |
| [32] | Martin, R. S.: Minimal positive harmonic functions. Math. Trans. Amer. Soc. 49(1), 137-172 (1941) · JFM 67.0343.03 · doi:10.1090/S0002-9947-1941-0003919-6 |
| [33] | Rao, M.: Hunt’s hypothesis for Lévy processes. Proc. Amer. Math. Soc. 104(2), 621-624 (1988) · Zbl 0693.60063 |
| [34] | Song, R., Wu, J.-M.: Boundary Harnack principle for symmetric stable processes. J. Funct. Anal. 168(2), 403-427 (1999) · Zbl 0945.31006 · doi:10.1006/jfan.1999.3470 |
| [35] | Wu, J. -M.: Comparisons of kernel functions, boundary Harnack principle and relative Fatou theorem on Lipschitz domains. Ann. Inst. Fourier 28(4), 147-167 (1978) · Zbl 0368.31006 · doi:10.5802/aif.719 |
| [36] | Yano, K.: Excursions away from a regular point for one-dimensional symmetric Lévy processes without Gaussian part. Potential Anal. 32, 305-341 (2010) · Zbl 1188.60023 · doi:10.1007/s11118-009-9152-6 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
