Abstract
Estimates of densities of convolution semigroups of probability measures are given under specific assumptions on the corresponding Lévy measure and the Lévy–Khinchin exponent. The assumptions are satisfied, e.g., by tempered stable semigroups of J. Rosiński.
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Bogdan, K., Jakubowski, T.: Estimates of heat kernel of fractional Laplacian perturbed by gradient operators. Commun. Math. Phys. 271(1), 179–198 (2007)
Bogdan, K., Sztonyk, P.: Estimates of potential kernel and Harnack’s inequality for anisotropic fractional Laplacian. Stud. Math. 181(2), 101–123 (2007)
Chen, Z.-Q., Kim, P., Kumagai, T.: Weighted Poincaré inequality and heat kernel estimates for finite range jump processes. Math. Ann. 342(4), 833–883 (2008)
Chen, Z.-Q., Kumagai, T.: Heat kernel estimates for stable-like processes on d-sets. Stoch. Process. Appl. 108(1), 27–62 (2003)
Chen, Z.-Q., Kumagai, T.: Heat kernel estimates for jump processes of mixed types on metric measure spaces. Probab. Theory Relat. Fields 140(1–2), 277–317 (2008)
Constantine, G.M., Savits, T.H.: A multivariate Faa di Bruno formula with applications. Trans. Am. Math. Soc. 348(2), 503–520 (1996)
Dziubański, J.: Asymptotic behaviour of densities of stable semigroups of measures. Probab. Theory Relat. Fields 87, 459–467 (1991)
Głowacki, P.: Lipschitz continuity of densities of stable semigroups of measures. Colloq. Math. 66(1), 29–47 (1993)
Głowacki, P., Hebisch, W.: Pointwise estimates for densities of stable semigroups of measures. Stud. Math. 104, 243–258 (1993)
Grzywny, T.: Potential theory for α-stable relativistic process. Master Thesis, Institut of Mathematics and Computer Sciences, Wrocław University of Technology (2005)
Grzywny, T., Ryznar, M.: Two-sided optimal bounds for Green functions of half-spaces for relativistic α-stable process. Potential Anal. 28(3), 201–239 (2008)
Hiraba, S.: Asymptotic behaviour of densities of multi-dimensional stable distributions. Tsukuba J. Math. 18(1), 223–246 (1994)
Hiraba, S.: Asymptotic estimates for densities of multi-dimensional stable distributions. Tsukuba J. Math. 27(2), 261–287 (2003)
Houdré, C., Kawai, R.: On layered stable processes. Bernoulli 13(1), 252–278 (2007)
Jacob, N.: Pseudo Differential Operators and Markov Processes. Fourier Analysis and Semigroups, vol. I. Imp. Coll. Press, London (2001)
Kulczycki, T., Siudeja, B.: Intrinsic ultracontractivity of the Feynman–Kac semigroup for relativistic stable processes. Trans. Am. Math. Soc. 358(11), 5025–5057 (2006)
Lewandowski, M.: Point regularity of p-stable density in ℛd and Fisher information. Probab. Math. Stat. 19(2), 375–388 (1999)
Picard, J.: Density in small time at accessible points for jump processes. Stochastic Process. Appl. 67(2), 251–279 (1997)
Pruitt, W.E., Taylor, S.J.: The potential kernel and hitting probabilities for the general stable process in R N. Trans. Am. Math. Soc. 146, 299–321 (1969)
Rosinski, J.: Tempering stable processes. Stoch. Process. Appl. 117(6), 677–707 (2007)
Ryznar, M.: Estimates of Green function for relativistic α-stable process. Potential Anal. 17(1), 1–23 (2002)
Sato, K.-I.: Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge (1999)
Sztonyk, P.: Regularity of harmonic functions for anisotropic fractional Laplacians. Math. Nach. (to appear)
Watanabe, T.: Asymptotic estimates of multi-dimensional stable densities and their applications. Trans. Am. Math. Soc. 359(6), 2851–2879 (2007)
Zaigraev, A.: On asymptotic properties of multidimensional α-stable densities. Math. Nachr. 279(16), 1835–1854 (2006)
Zolotarev, V.M.: One-Dimensional Stable Distributions. Am. Math. Soc., Providence (1986)
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The research was supported by The State Committee for Scientific Research (Poland, KBN 1 P03A 026 29) and The Alexander von Humboldt Foundation (Germany, 3-PL/1122470).
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Sztonyk, P. Estimates of Tempered Stable Densities. J Theor Probab 23, 127–147 (2010). https://doi.org/10.1007/s10959-009-0208-8
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DOI: https://doi.org/10.1007/s10959-009-0208-8