Scale invariant boundary Harnack principle at infinity for Feller processes. (English) Zbl 1381.60110
The boundary Harnack principle is a result expressing that nonnegative functions, which are harmonic in an open set and vanish near a part of the boundary of the open set, have the same boundary decay rate near that part of boundary. In the present paper, a uniform and scale boundary Harnack principle at infinity is proved for a large class of processes.
Reviewer: Sophia L. Kalpazidou (Thessaloniki)
MSC:
| 60J50 | Boundary theory for Markov processes |
| 31C40 | Fine potential theory; fine properties of sets and functions |
| 31C35 | Martin boundary theory |
| 60J45 | Probabilistic potential theory |
| 60J75 | Jump processes (MSC2010) |
References:
| [1] | Ancona, A.: Principe de Harnack à la frontière et théorème de Fatou pour un opérateur elliptique dans un domaine lipschitzien. Ann. Inst. Fourier (Grenoble) 28, 169-213 (1978) · Zbl 0377.31001 · doi:10.5802/aif.720 |
| [2] | Blumenthal, R. M., Getoor, R. K.: Markov processes and potential theory. Academic Press (1968) · Zbl 0169.49204 |
| [3] | Bogdan, K.: The boundary Harnack principle for the fractional Laplacian. Studia Math. 123, 43-80 (1997) · Zbl 0870.31009 · doi:10.4064/sm-123-1-43-80 |
| [4] | Bogdan, K., Grzywny, T., Ryznar, M.: Density and tails of unimodal convolution semigroups. J. Funct. Anal. 266, 3543-3571 (2014) · Zbl 1290.60002 · doi:10.1016/j.jfa.2014.01.007 |
| [5] | Bogdan, K., Kulczycki, T., Kwaśnicki, M.: Estimates and structure of α-harmonic functions. Probab. Theory Relat. Fields 140, 345-381 (2008) · Zbl 1146.31004 · doi:10.1007/s00440-007-0067-0 |
| [6] | 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 |
| [7] | 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 |
| [8] | Chen, Z. -Q., Kumagai, T.: Heat kernel estimates for jump processes of mixed types on metric measure spaces. Probab. Theory Relat. Fields 140, 277-317 (2008) · Zbl 1131.60076 · doi:10.1007/s00440-007-0070-5 |
| [9] | Chung, K. L.: Doubly-Feller Process with Multiplicative Functional, Seminar on Stochastic Processes, 1985, 63-78, Progr. Probab. Statist., vol. 12. Birkhäuser Boston, Boston, MA (1986) · Zbl 0368.31006 |
| [10] | Dahlberg, B. E. J.: Estimates of harmonic measure. Arch. Ration. Mech. Anal. 65, 275-288 (1977) · Zbl 0406.28009 · doi:10.1007/BF00280445 |
| [11] | Grzywny, T.: On Harnack inequality and hölder regularity for isotropic unimodal lévy processes. Potential Anal. 41, 1-29 (2014) · Zbl 1302.60109 · doi:10.1007/s11118-013-9360-y |
| [12] | Kim, I., Kim, K. -H., Kim, P.: Parabolic Littlewood-Paley inequality for Φ(−Δ)-type operators and applications to stochastic integro-differential equations. Adv. Math. 249, 161-203 (2013) · Zbl 1287.42015 · doi:10.1016/j.aim.2013.09.008 |
| [13] | Kim, P., Mimica, A.: Green function estimates for subordinate Brownian motions: stable and beyond. Trans. Amer. Math. Soc. 366, 4383-4422 (2014) · Zbl 1327.60165 · doi:10.1090/S0002-9947-2014-06017-0 |
| [14] | Kim, P., Song, R., Vondraček, Z.: Boundary Harnack principle for subordinate Brownian motions. Stoch. Process. Appl. 119, 1601-1631 (2009) · Zbl 1166.60046 · doi:10.1016/j.spa.2008.08.003 |
| [15] | Kim, P., Song, R., Vondraček, Z.: Potential Theory of Subordinate Brownian Motions Revisited, Stochastic Analysis and Applications to Finance: Essays in Honour of Jia-an Yan, 243-290, Interdiscip. Math Sci., vol. 13. World Sci. Publ., Hackensack, NJ (2012) · Zbl 1282.60076 |
| [16] | Kim, P., Song, R., Vondraček, Z.: Two-sided Green function estimates for killed subordinate Brownian motions. Proc. London Math. Soc. 104, 927-958 (2012) · Zbl 1248.60091 · doi:10.1112/plms/pdr050 |
| [17] | Kim, P., Song, R., Vondraček, Z.: Uniform boundary Harnack principle for rotationally symmetric lévy processes in general open sets, Sci. China Math. 55, 2317-2333 (2012) · Zbl 1262.60077 · doi:10.1007/s11425-012-4516-6 |
| [18] | Kim, P., Song, R., Vondraček, Z.: Potential theory of subordinate Brownian motions with Gaussian components. Stoch. Process. Appl. 123, 764-795 (2013) · Zbl 1266.31007 · doi:10.1016/j.spa.2012.11.007 |
| [19] | Kim, P., Song, R., Vondraček, Z.: Global uniform boundary Harnack principle with explicit decay rate and its application. Stoch. Process. Appl. 124, 235-267 (2014) · Zbl 1296.60198 · doi:10.1016/j.spa.2013.07.007 |
| [20] | Kim, P., Song, R., Vondraček, Z.: Boundary Harnack principle and Martin boundary at infinity for subordinate Brownian motion. Potential Anal. 41, 407-441 (2014) · Zbl 1302.60110 · doi:10.1007/s11118-013-9375-4 |
| [21] | Kim, P., Song, R., Vondraček, Z.: Martin boundary for some symmetric Lévy processes. In: Festschrift Masatoshi Fukushima: In honor of Masatoshi Fukushima’s Sanju. Interdisciplinary Mathematical Sciences, vol. 17, pp 307-342. World Scientific, Hackensack, NJ (2015) · Zbl 1336.31031 |
| [22] | 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 (2017) · Zbl 1395.60094 |
| [23] | Kwaśnicki, M.: Intrinsic ultracontractivity for stable semigroups on unbounded open sets. Potential Anal. 31, 57-77 (2009) · Zbl 1180.47031 · doi:10.1007/s11118-009-9125-9 |
| [24] | Rao, M.: Hunt’s hypothesis for lévy processes. Proc. Amer. Math. Soc. 104, 621-624 (1988) · Zbl 0693.60063 |
| [25] | Schilling, R. L., Wang, J.: Strong Feller continuity of Feller processes and semigroups. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 15(2), 1250010, 28 (2012) · Zbl 1258.60047 · doi:10.1142/S0219025712500105 |
| [26] | Song, R., Wu, J. -M.: Boundary Harnack principle for symmetric stable processes. J. Funct. Anal. 168, 403-427 (1999) · Zbl 0945.31006 · doi:10.1006/jfan.1999.3470 |
| [27] | Vondraček, Z.: Basic potential theory of certain nonsymmetric strictly α-stable processes. Glas. Mat. 37, 211-233 (2002) · Zbl 1009.60063 |
| [28] | Wu, J. -M.: Comparisons of kernel functions, boundary Harnack principle and relative Fatou theorem on Lipschitz domains. Ann. Inst. Fourier (Grenoble) 28, 147-167 (1978) · Zbl 0368.31006 · doi:10.5802/aif.719 |
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.
