Non-symmetric stable processes: Dirichlet heat kernel, Martin kernel and Yaglom limit. (English) Zbl 1543.60054
Summary: We study a \(d\)-dimensional non-symmetric strictly \(\alpha\)-stable Lévy process \(\mathbf{X}\), whose spherical density is bounded and bounded away from the origin. First, we give sharp two-sided estimates on the transition density of \(\mathbf{X}\) killed when leaving an arbitrary \(\kappa\)-fat set. We apply these results to get the existence of the Yaglom limit for arbitrary \(\kappa\)-fat cone. In the meantime we also obtain the spacial asymptotics of the survival probability at the vertex of the cone expressed by means of the Martin kernel for \(\varGamma\) and its homogeneity exponent. Our results hold for the dual process \(\widehat{\mathbf{X}}\), too.
MSC:
| 60G52 | Stable stochastic processes |
| 60G51 | Processes with independent increments; Lévy processes |
| 60J35 | Transition functions, generators and resolvents |
Keywords:
non-symmetric stable process; heat kernel estimates; Dirichlet problem; Yaglom limit; Martin kernelReferences:
| [1] | Armstrong, G.; Bogdan, K.; Grzywny, T.; Leżaj, Ł.; Wang, L., Yaglom limit for unimodal Lévy processes, Ann. Inst. Henri Poincaré Probab. Stat., 59, 3, 1688-1721, 2023 · Zbl 07788719 |
| [2] | Asselah, A.; Ferrari, P. A.; Groisman, P.; Jonckheere, M., Fleming-Viot selects the minimal quasi-stationary distribution: The galton-watson case, Ann. Inst. Henri Poincaré Probab. Stat., 52, 2, 647-668, 2016 · Zbl 1342.60145 |
| [3] | Bañuelos, R.; Bogdan, K., Symmetric stable processes in cones, Potential Anal., 21, 3, 263-288, 2004 · Zbl 1054.31002 |
| [4] | Bass, R. F.; Levin, D. A., Harnack inequalities for jump processes, Potential Anal., 17, 4, 375-388, 2002 · Zbl 0997.60089 |
| [5] | Bean, N.; Bright, L.; Latouche, G.; Pearce, C.; Pollett, P.; Taylor, P., The quasi-stationary behavior of quasi-birth-and-death processes, Ann. Appl. Probab., 7, 1, 134-155, 1997 · Zbl 0883.60085 |
| [6] | Bertoin, J.; Doney, R. A., On conditioning a random walk to stay nonnegative, Ann. Probab., 22, 4, 2152-2167, 1994 · Zbl 0834.60079 |
| [7] | Blumenthal, R. M.; Getoor, R. K., Markov processes and potential theory, (Pure and Applied Mathematics, Vol. 29, 1968, Academic Press: Academic Press New York-London), x+313 · Zbl 0169.49204 |
| [8] | Bogdan, K., The boundary harnack principle for the fractional Laplacian, Studia Math., 123, 1, 43-80, 1997 · Zbl 0870.31009 |
| [9] | Bogdan, K., Sharp estimates for the green function in Lipschitz domains, J. Math. Anal. Appl., 243, 2, 326-337, 2000 · Zbl 0971.31005 |
| [10] | Bogdan, K.; Grzywny, T.; Ryznar, M., Heat kernel estimates for the fractional Laplacian with Dirichlet, conditions, Ann. Probab., 38, 5, 1901-1923, 2010 · Zbl 1204.60074 |
| [11] | Bogdan, K.; Grzywny, T.; Ryznar, M., Dirichlet heat kernel for unimodal Lévy processes, Stochastic Process. Appl., 124, 11, 3612-3650, 2014 · Zbl 1320.60112 |
| [12] | Bogdan, K.; Jakubowski, T.; Kim, P.; Pilarczyk, D., Self-similar solution for Hardy operator, J. Funct. Anal., 285, 5, 2023, Paper No. 110014, 40 · Zbl 07694902 |
| [13] | Bogdan, K.; Knosalla, P.; Leżaj, Ł.; Pilarczyk, D., Self-similar solution for fractional Laplacian in cones, Electron. J. Probab., 29, 1-24, 2024, article no. 54 · Zbl 1534.60047 |
| [14] | Bogdan, K.; Kulczycki, T.; Kwaśnicki, M., Estimates and structure of \(\alpha \)-harmonic functions, Probab. Theory Related Fields, 140, 3-4, 345-381, 2008 · Zbl 1146.31004 |
| [15] | Bogdan, K.; Kumagai, T.; Kwaśnicki, M., Boundary harnack inequality for Markov processes with jumps, Trans. Amer. Math. Soc., 367, 1, 477-517, 2015 · Zbl 1309.60080 |
| [16] | Bogdan, K.; Palmowski, Z.; Wang, L., Yaglom limit for stable processes in cones, Electron. J. Probab., 23, 1-19, 2018 · Zbl 1390.31002 |
| [17] | Chaumont, L., Conditionings and path decompositions for Lévy processes, Stochastic Process. Appl., 64, 1, 39-54, 1996 · Zbl 0879.60072 |
| [18] | Chaumont, L.; Doney, R. A., On Lévy processes conditioned to stay positive, Electron. J. Probab., 10, 28, 948-961, 2005 · Zbl 1109.60039 |
| [19] | Chen, Z.-Q.; Hu, E.; Zhao, G., Dirichlet heat kernel estimates for cylindrical stable processes, 2023, Preprint. arXiv:2304.14026 |
| [20] | Chen, Z.-Q.; Kim, P.; Song, R., Dirichlet heat kernel estimates for \(\Delta^{\alpha / 2} + \Delta^{\beta / 2}\), Illinois J. Math., 54, 4, 1357-1392, 2010 · Zbl 1268.60101 |
| [21] | Chen, Z.-Q.; Kim, P.; Song, R., Heat kernel estimates for the Dirichlet fractional Laplacian, J. Eur. Math. Soc. (JEMS), 12, 5, 1307-1329, 2010 · Zbl 1203.60114 |
| [22] | Chen, Z.-Q.; Kim, P.; Song, R., Sharp heat kernel estimates for relativistic stable processes in open sets, Ann. Probab., 40, 1, 213-244, 2012 · Zbl 1235.60101 |
| [23] | Chen, Z.-Q.; Kim, P.; Song, R., Dirichlet heat kernel estimates for rotationally symmetric Lévy processes, Proc. Lond. Math. Soc., 109, 1, 90-120, 2014 · Zbl 1304.60080 |
| [24] | Chen, Z.-Q.; Song, R., Estimates on green functions and Poisson kernels for symmetric stable processes, Math. Ann., 312, 3, 465-501, 1998 · Zbl 0918.60068 |
| [25] | K.L. Chung, J.B. Walsh, Markov processes, Brownian motion, and time symmetry, second ed., in: Grundlehren der mathematischen Wissenschaften[Fundamental Principles of Mathematical Sciences], vol. 249, Springer, New York, ISBN: 978-0387-22026-0; 0-387-22026-7, 2005, p. xii+431. · Zbl 1082.60001 |
| [26] | Chung, K. L.; Zhao, Z. X., From Brownian motion to Schrödinger’s equation, xii+287, 1995, Springer-Verlag: Springer-Verlag Berlin · Zbl 0819.60068 |
| [27] | Dipierro, S.; Ros-Oton, X.; Serra, J.; Valdinoci, E., Non-symmetric stable operators: regularity theory and integration by parts, Adv. Math., 401, 2022, Paper No. 108321, 100 · Zbl 1490.45011 |
| [28] | Ferrari, P. A.; Kesten, H.; Martinez, S.; Picco, P., Existence of quasi-stationary distributions. a renewal dynamical approach, Ann. Probab., 23, 2, 501-521, 1995 · Zbl 0827.60061 |
| [29] | Flaspohler, D. C.; Holmes, P. T., Additional quasi-stationary distributions for semi-Markov processes, J. Appl. Probab., 9, 671-676, 1972 · Zbl 0241.60080 |
| [30] | Grzywny, T.; Kim, K.-Y.; Kim, P., Estimates of Dirichlet heat kernel for symmetric Markov processes, Stochastic Process. Appl., 130, 1, 431-470, 2020 · Zbl 1468.60097 |
| [31] | Grzywny, T.; Leżaj, Ł.; Miśta, M., Hitting probabilities for Lévy processes on the real line, ALEA, Lat. Am. J. Probab. Math. Stat., 18, 727-760, 2021 · Zbl 1469.60146 |
| [32] | Haas, B.; Rivero, V., Quasi-stationary distributions and yaglom limits of self-similar Markov processes, Stochastic Process. Appl., 122, 12, 4054-4095, 2012 · Zbl 1267.60038 |
| [33] | Harris, S. C.; Horton, E.; Kyprianou, A. E.; Wang, M., Yaglom limit for critical nonlocal branching Markov processes, Ann. Probab., 50, 6, 2373-2408, 2022 · Zbl 1505.82063 |
| [34] | Hartman, P.; Wintner, A., On the infinitesimal generators of integral convolutions, Amer. J. Math., 64, 273-298, 1942 · Zbl 0063.01951 |
| [35] | Iglehart, D. L., Random walks with negative drift conditioned to stay positive, J. Appl. Probab., 11, 742-751, 1974 · Zbl 0302.60038 |
| [36] | Ikeda, N.; Watanabe, S., On some relations between the harmonic measure and the Lévy measure for a certain class of Markov processes, J. Math. Kyoto Univ., 2, 79-95, 1962 · Zbl 0118.13401 |
| [37] | Jacka, S. D.; Roberts, G. O., Weak convergence of conditioned processes on a countable state space, J. Appl. Probab., 32, 4, 902-916, 1995 · Zbl 0839.60069 |
| [38] | Jakubowski, T., The estimates for the green function in Lipschitz domains for the symmetric stable processes, Probab. Math. Statist., 22, 2, 419-441, 2002, Acta Univ. Wratislav. No. 2470 · Zbl 1035.60046 |
| [39] | Juszczyszyn, T., Decay rate of harmonic functions for non-symmetric strictly \(\alpha \)-stable Lévy processes, Studia Math., 260, 2, 141-165, 2021 · Zbl 1479.60097 |
| [40] | Kim, P.; Mimica, A., Estimates of Dirichlet heat kernels for subordinate Brownian motions, Electron. J. Probab., 23, 2018, Paper No. 64, 45 · Zbl 1410.60082 |
| [41] | Kim, P.; Song, R.; Vondraček, Z., Scale invariant boundary harnack principle at infinity for feller processes, Potential Anal., 47, 3, 337-367, 2017 · Zbl 1381.60110 |
| [42] | Kim, P.; Song, R.; Vondraček, Z., Accessibility, martin boundary and minimal thinness for feller processes in metric measure spaces, Rev. Mat. Iberoam., 34, 2, 541-592, 2018 · Zbl 1395.60094 |
| [43] | Kulczycki, T., Properties of green function of symmetric stable processes, Probab. Math. Statist., 17, 2, Acta Univ. Wratislav. No. 2029, 339-364, 1997 · Zbl 0903.60063 |
| [44] | Kulczycki, T., Exit time and green function of cone for symmetric stable processes, Probab. Math. Statist., 19, 2, Acta Univ. Wratislav. No. 2198, 337-374, 1999 · Zbl 0986.60071 |
| [45] | Kulczycki, T.; Ryznar, M., Gradient estimates of harmonic functions and transition densities for Lévy processes, Trans. Amer. Math. Soc., 368, 1, 281-318, 2016 · Zbl 1336.31012 |
| [46] | Kulczycki, T.; Siudeja, B., Intrinsic ultracontractivity of the feynman-kac semigroup for relativistic stable processes, Trans. Amer. Math. Soc., 358, 11, 5025-5057, 2006 · Zbl 1112.47034 |
| [47] | Kulik, A.; Scheutzow, M., A coupling approach to doob’s theorem, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 26, 1, 83-92, 2015 · Zbl 1310.60104 |
| [48] | Kyprianou, A. E.; Palmowski, Z., Quasi-stationary distributions for Lévy processes, Bernoulli, 12, 4, 571-581, 2006 · Zbl 1130.60054 |
| [49] | Kyprianou, A. E.; Rivero, V.; Satitkanitkul, W., Stable Lévy processes in a cone, Ann. Inst. Henri Poincaré Probab. Stat., 57, 4, 2066-2099, 2021 · Zbl 1489.60080 |
| [50] | Kyprianou, A. E.; Rivero, V.; Şengül, B.; Yang, T., Entrance laws at the origin of self-similar Markov processes in high dimensions, Trans. Amer. Math. Soc., 373, 9, 6227-6299, 2020 · Zbl 1469.60111 |
| [51] | Lambert, A., Quasi-stationary distributions and the continuous-state branching process conditioned to be never extinct, Electron. J. Probab., 12, 14, 420-446, 2007 · Zbl 1127.60082 |
| [52] | Mandjes, M.; Palmowski, Z.; Rolski, T., Quasi-stationary workload in a Lévy-driven storage system, Stoch. Models, 28, 3, 413-432, 2012 · Zbl 1260.90020 |
| [53] | Martínez, S.; San Martín, J., Quasi-stationary distributions for a Brownian motion with drift and associated limit laws, J. Appl. Probab., 31, 4, 911-920, 1994 · Zbl 0818.60071 |
| [54] | Michalik, K., Sharp estimates of the green function, the Poisson kernel and the martin kernel of cones for symmetric stable processes, Hiroshima Math. J., 36, 1, 1-21, 2006 · Zbl 1103.31003 |
| [55] | Millar, P. W., First passage distributions of processes with independent increments, Ann. Probability, 3, 215-233, 1975 · Zbl 0318.60063 |
| [56] | Palmowski, Z.; Vlasiou, M., Speed of convergence to the quasi-stationary distribution for Lévy input fluid queues, Queueing Syst., 96, 1-2, 153-167, 2020 · Zbl 1461.60031 |
| [57] | Profeta, C.; Simon, T., On the harmonic measure of stable processes, (SÉMinaire de ProbabilitÉS XLVIII, 2016, Springer International Publishing), 325-345 · Zbl 1367.60057 |
| [58] | Pruitt, W. E., The growth of random walks and Lévy processes, Ann. Probab., 9, 6, 948-956, 1981 · Zbl 0477.60033 |
| [59] | Sato, K.-i., Lévy processes and infinitely divisible distributions, Cambridge Studies in Advanced Mathematics, xii+486, 1999, Cambridge University Press: Cambridge University Press Cambridge, Translated from the 1990 Japanese original, Revised by the author · Zbl 0973.60001 |
| [60] | Seneta, E.; Vere-Jones, D., On quasi-stationary distributions in discrete-time Markov chains with a denumerable infinity of states, J. Appl. Probab., 3, 403-434, 1966 · Zbl 0147.36603 |
| [61] | Siudeja, B., Symmetric stable processes on unbounded domains, Potential Anal., 25, 4, 371-386, 2006 · Zbl 1111.31004 |
| [62] | Song, R.; Vondraček, Z., Harnack inequality for some classes of Markov processes, Math. Z., 246, 1-2, 177-202, 2004 · Zbl 1052.60064 |
| [63] | Song, R.; Wu, J.-M., Boundary harnack principle for symmetric stable processes, J. Funct. Anal., 168, 2, 403-427, 1999 · Zbl 0945.31006 |
| [64] | Sztonyk, P., On harmonic measure for Lévy processes, Probab. Math. Statist., 20, 2, Acta Univ. Wratislav. No. 2256, 383-390, 2000 · Zbl 0991.60067 |
| [65] | Taylor, S. J., Sample path properties of a transient stable process, J. Math. Mech., 16, 1229-1246, 1967 · Zbl 0178.19301 |
| [66] | Tweedie, R. L., Quasi-stationary distributions for Markov chains on a general state space, J. Appl. Probab., 11, 726-741, 1974 · Zbl 0309.60040 |
| [67] | van Doorn, E. A., Quasi-stationary distributions and convergence to quasi-stationarity of birth-death processes, Adv. in Appl. Probab., 23, 4, 683-700, 1991, 1133722 · Zbl 0736.60076 |
| [68] | van Doorn, E. A.; Pollett, P. K., Quasi-stationary distributions for discrete-state models, European J. Oper. Res., 230, 1, 1-14, 2013 · Zbl 1317.60093 |
| [69] | Varopoulos, N. T., Gaussian estimates in Lipschitz domains, Canad. J. Math., 55, 2, 401-431, 2003 · Zbl 1042.58013 |
| [70] | Vondraček, Z., Basic potential theory of certain nonsymmetric strictly \(\alpha \)-stable processes, Glas. Mat. Ser. III, 37, 57, 211-233, 2002 · Zbl 1009.60063 |
| [71] | Watanabe, T., Asymptotic estimates of multi-dimensional stable densities and their applications, Trans. Amer. Math. Soc., 359, 6, 2851-2879, 2007 · Zbl 1124.62063 |
| [72] | Yaglom, A. M., Certain limit theorems of the theory of branching random processes, Dokl. Akad. Nauk SSSR, 56, 795-798, 1947 · Zbl 0041.45602 |
| [73] | Zhang, Q. S., The boundary behavior of heat kernels of Dirichlet Laplacians, J. Differential Equations, 182, 2, 416-430, 2002 · Zbl 1002.35052 |
| [74] | Zhang, J.; Li, S.; Song, R., Quasi-stationarity and quasi-ergodicity of general Markov processes, Sci. China Math., 57, 10, 2013-2024, 2014 · Zbl 1309.60078 |
| [75] | Zhao, Z. X., Green function for Schrödinger operator and conditioned feynman-kac gauge, J. Math. Anal. Appl., 116, 2, 309-334, 1986 · Zbl 0608.35012 |
| [76] | Zolotarev, V. M., Mellin-stieltjes transformations in probability theory, Teor. Veroyatn. Primen., 2, 444-469, 1957 · Zbl 0084.14103 |
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