Estimates of transition densities and their derivatives for jump Lévy processes. (English) Zbl 1317.60056
Summary: We give upper and lower estimates of densities of convolution semigroups of probability measures under explicit assumptions on the corresponding Lévy measure and the Lévy-Khinchin exponent. We obtain also estimates of derivatives of densities.
MSC:
| 60G51 | Processes with independent increments; Lévy processes |
| 60J75 | Jump processes (MSC2010) |
| 60G52 | Stable stochastic processes |
| 35K08 | Heat kernel |
Keywords:
jump Lévy processes; transition densities; stable processes; semigroup of measures; heat kernelReferences:
| [1] | Blumenthal, R. M.; Getoor, R. K., Some theorems on stable processes, Trans. Amer. Math. Soc., 95, 263-273 (1960) · Zbl 0107.12401 |
| [2] | Bogdan, K.; Grzywny, T.; Ryznar, M., Density and tails of unimodal convolution semigroups, J. Funct. Anal., 266, 6, 3543-3571 (2014) · Zbl 1290.60002 |
| [3] | Bogdan, K.; Jakubowski, T., Estimates of heat kernel of fractional Laplacian perturbed by gradient operators, Comm. Math. Phys., 271, 1, 179-198 (2007) · Zbl 1129.47033 |
| [4] | Bogdan, K.; Sztonyk, P., Estimates of potential kernel and Harnack’s inequality for anisotropic fractional Laplacian, Studia Math.. Studia Math., J. Funct. Anal., 91, 2, 117-142 (1990) |
| [5] | Chen, Z.-Q.; Kim, P.; Kumagai, T., Global heat kernel estimates for symmetric jump processes, Trans. Amer. Math. Soc., 363, 9, 5021-5055 (2011) · Zbl 1234.60088 |
| [6] | Chen, Z.-Q.; Kumagai, T., Heat kernel estimates for jump processes of mixed types on metric measure spaces, Probab. Theory Related Fields, 140, 1-2, 277-317 (2008) · Zbl 1131.60076 |
| [7] | Dziubański, J., Asymptotic behaviour of densities of stable semigroups of measures, Probab. Theory Related Fields, 87, 459-467 (1991) · Zbl 0695.60013 |
| [8] | Głowacki, P., Lipschitz continuity of densities of stable semigroups of measures, Colloq. Math., 66, 1, 29-47 (1993) · Zbl 0837.43009 |
| [9] | Głowacki, P.; Hebisch, W., Pointwise estimates for densities of stable semigroups of measures, Studia Math., 104, 243-258 (1993) · Zbl 0812.43005 |
| [10] | Grzywny, T., On Harnack inequality and Hölder regularity for isotropic unimodal Lévy processes, Potential Anal., 41, 1-29 (2014) · Zbl 1302.60109 |
| [11] | Hiraba, S., Asymptotic behaviour of densities of multi-dimensional stable distributions, Tsukuba J. Math., 18, 1, 223-246 (1994) · Zbl 0807.60021 |
| [12] | Hiraba, S., Asymptotic estimates for densities of multi-dimensional stable distributions, Tsukuba J. Math., 27, 2, 261-287 (2003) · Zbl 1172.62307 |
| [13] | Jacob, N., Pseudo Differential Operators and Markov Processes, vol. I: Fourier Analysis and Semigroups (2001), Imperial College Press: Imperial College Press London · Zbl 0987.60003 |
| [14] | Jacob, N.; Knopova, V.; Landwehr, S.; Schilling, R., A geometric interpretation of the transition density of a symmetric Lévy process, Sci. China Math., 55, 6, 1099-1126 (2012) · Zbl 1279.60097 |
| [15] | Kaleta, K.; Sztonyk, P., Upper estimates of transition densities for stable-dominated semigroups, J. Evol. Equ., 13, 3, 633-650 (2013) · Zbl 1298.60086 |
| [16] | Knopova, V.; Kulik, A., Exact asymptotic for distribution densities of Lévy functionals, Electron. J. Probab., 16, 1394-1433 (2011) · Zbl 1245.60051 |
| [17] | Knopova, V.; Kulik, A., Intrinsic small time estimates for distribution densities of Lévy processes, Random Oper. Stoch. Equ., 21, 4, 321-344 (2013) · Zbl 1291.60095 |
| [18] | Knopova, V.; Schilling, R., Transition density estimates for a class of Lévy and Lévy-type processes, J. Theoret. Probab., 25, 1, 144-170 (2012) · Zbl 1242.60082 |
| [19] | Knopova, V.; Schilling, R., A note on the existence of transition probability densities for Lévy processes, Forum Math., 25, 1, 125-149 (2013) · Zbl 1269.60050 |
| [20] | Mimica, A., Heat kernel upper estimates for symmetric jump processes with small jumps of high intensity, Potential Anal., 36, 2, 203-222 (2012) · Zbl 1239.60077 |
| [21] | Mimica, A., Harnack inequality and Hölder regularity estimates for a Lévy process with small jumps of high intensity, J. Theoret. Probab., 26, 2, 329-348 (2013) · Zbl 1279.60100 |
| [22] | Picard, J., On the existence of small densities for jump processes, Probab. Theory Related Fields, 105, 481-511 (1996) · Zbl 0853.60064 |
| [23] | Picard, J., Density in small time at accessible points for jump processes, Stochastic Process. Appl., 67, 2, 251-279 (1997) · Zbl 0889.60088 |
| [24] | Pruitt, W. E., The growth of random walks and Lévy processes, Ann. Probab., 9, 6, 948-956 (1981) · Zbl 0477.60033 |
| [25] | Pruitt, W. E.; Taylor, S. J., The potential kernel and hitting probabilities for the general stable process in \(R^N\), Trans. Amer. Math. Soc., 146, 299-321 (1969) · Zbl 0229.60052 |
| [26] | Sato, K.-I., Lévy Processes and Infinitely Divisible Distributions (1999), Cambridge University Press · Zbl 0973.60001 |
| [27] | Schilling, R., Growth and Hölder conditions for the sample paths of Feller processes, Probab. Theory Related Fields, 112, 4, 565-611 (1998) · Zbl 0930.60013 |
| [28] | Schilling, R.; Sztonyk, P.; Wang, J., Coupling property and gradient estimates for Lévy processes via the symbol, Bernoulli, 18, 1128-1149 (2012) · Zbl 1263.60045 |
| [29] | Sztonyk, P., Regularity of harmonic functions for anisotropic fractional Laplacians, Math. Nachr., 283, 2, 289-311 (2010) · Zbl 1194.47044 |
| [30] | Sztonyk, P., Estimates of tempered stable densities, J. Theoret. Probab., 23, 1, 127-147 (2010) · Zbl 1393.60050 |
| [31] | Sztonyk, P., Transition density estimates for jump Lévy processes, Stochastic Process. Appl., 121, 1245-1265 (2011) · Zbl 1217.60036 |
| [32] | Watanabe, T., Asymptotic estimates of multi-dimensional stable densities and their applications, Trans. Amer. Math. Soc., 359, 6, 2851-2879 (2007) · Zbl 1124.62063 |
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.
