Summary
We prove that densities of the measures in a strictly stable semigroup (h t ) of symmetric measures on a homogeneous group, if they exist, have the following asymptotic behaviour:
where α is the characteristic exponent,\(\bar x = |x|^{ - 1} x\), andk is the density of the Lévy measure associated to the semigroup. Moreover, if\(k(\bar x) = 0\) a more precise description is given.
References
Bourbaki, N.: Groupes et algèbres de Lie, Chap. I–III. (Elements de Math, Fasc. 26 et 37) Paris: Hermann 1960 and 1972
Duflo, M.: Representations de semi-groupes de mesures sur un groupe localment compact. Ann. Inst. Fourier28, 225–249 (1978)
Feller, W.: An introduction to probability theory and its application II. New York: Wiley 1966
Folland, G.B., Stein, E.M.: Hardy spaces on homogeneous groups. Princeton, N.J.: Princeton University Press 1982
Gikhman, I.I., Skorohod, A.W.: Introduction to stochastic processes theory (in Russian). Moscow: Nauka 1965
Głowacki, P.: Stable semigroups of measures as commutative approximate identities on non-graded homogeneous groups. Invent. Math.83, 557–582 (1986)
Głowacki, P.: The Rockland condition for non-differential convolution operators. Duke Math. J.58, 371–395 (1989)
Głowacki, P., Hebisch, W.: Pointwise estimates for densities of stable semigroups of measures. Preprint, Wrocław 1990
Hulanicki, A.: A class of convolution semigroups of measures on a Lie group. (Lect. Notes Math., vol. 828, pp. 82–101) Berlin Heidelberg New York: Springer 1980
Hunt, G.A.: Semi-groups of measures on Lie groups. Trans. Am. Math. Soc.81, 264–293 (1956)
Janssen, A.: Charakterisierung stetiger Faltungshalbgruppen durch das Lévy-Maß. Math. Ann.246, 233–240 (1980)
Pazy, P.: Semigroups of linear operators and applications to partial differential equations. Berlin Heidelberg New York: Springer 1983
Schwartz, L.: Théorie des distributions. Paris: Hermann 1966
Sharpe, M.: Operator-stable probability distribution on vector groups. Trans. Am. Math. Soc.136, 51–65 (1969)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Dziubański, J. Asymptotic behaviour of densities of stable semigroups of measures. Probab. Th. Rel. Fields 87, 459–467 (1991). https://doi.org/10.1007/BF01304275
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01304275
Keywords
- Stochastic Process
- Asymptotic Behaviour
- Probability Theory
- Mathematical Biology
- Homogeneous Group