Probability Theory and Related Fields

, Volume 87, Issue 4, pp 459–467 | Cite as

Asymptotic behaviour of densities of stable semigroups of measures

  • Jacek Dziubański


We prove that densities of the measures in a strictly stable semigroup (ht) of symmetric measures on a homogeneous group, if they exist, have the following asymptotic behaviour:
$$\mathop {\lim |}\limits_{|x| \to \infty } x|^{Q + \alpha } \cdot h_1 (x) = k(\bar x),$$
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.


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  1. 1.
    Bourbaki, N.: Groupes et algèbres de Lie, Chap. I–III. (Elements de Math, Fasc. 26 et 37) Paris: Hermann 1960 and 1972Google Scholar
  2. 2.
    Duflo, M.: Representations de semi-groupes de mesures sur un groupe localment compact. Ann. Inst. Fourier28, 225–249 (1978)Google Scholar
  3. 3.
    Feller, W.: An introduction to probability theory and its application II. New York: Wiley 1966Google Scholar
  4. 4.
    Folland, G.B., Stein, E.M.: Hardy spaces on homogeneous groups. Princeton, N.J.: Princeton University Press 1982Google Scholar
  5. 5.
    Gikhman, I.I., Skorohod, A.W.: Introduction to stochastic processes theory (in Russian). Moscow: Nauka 1965Google Scholar
  6. 6.
    Głowacki, P.: Stable semigroups of measures as commutative approximate identities on non-graded homogeneous groups. Invent. Math.83, 557–582 (1986)Google Scholar
  7. 7.
    Głowacki, P.: The Rockland condition for non-differential convolution operators. Duke Math. J.58, 371–395 (1989)Google Scholar
  8. 8.
    Głowacki, P., Hebisch, W.: Pointwise estimates for densities of stable semigroups of measures. Preprint, Wrocław 1990Google Scholar
  9. 9.
    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 1980Google Scholar
  10. 10.
    Hunt, G.A.: Semi-groups of measures on Lie groups. Trans. Am. Math. Soc.81, 264–293 (1956)Google Scholar
  11. 11.
    Janssen, A.: Charakterisierung stetiger Faltungshalbgruppen durch das Lévy-Maß. Math. Ann.246, 233–240 (1980)Google Scholar
  12. 12.
    Pazy, P.: Semigroups of linear operators and applications to partial differential equations. Berlin Heidelberg New York: Springer 1983Google Scholar
  13. 13.
    Schwartz, L.: Théorie des distributions. Paris: Hermann 1966Google Scholar
  14. 14.
    Sharpe, M.: Operator-stable probability distribution on vector groups. Trans. Am. Math. Soc.136, 51–65 (1969)Google Scholar

Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • Jacek Dziubański
    • 1
  1. 1.Institute of Mathematics Polish Academy of SciencesMathematical Institute University of WrocławWrocławPoland

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