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Title

Malliavin calculus for stable processes on homogeneous groups

Authors 1

Affiliations

  1. Institute of Mathematics, Wrocław Technical University, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland

Abstract

Let {μt}t>0 be a symmetric semigroup of stable measures on a homogeneous group, with smooth Lévy measure. Applying Malliavin calculus for jump processes we prove that the measures μt have smooth densities.

Bibliography

  1. J.-M. Bismut, Calcul des variations stochastique et processus de sauts, Z. Wahrsch. Verw. Gebiete 63 (1983), 147-235.
  2. J.-M. Bismut, Jump processes and boundary processes, in: Stochastic Analysis, Proc. Taniguchi Internat. Sympos., Katata and Kyoto, 1982, K. Itô (ed.), North-Holland Math. Library 32, Kinokuniya and North-Holland, 1984, 53-104.
  3. S. Ethier and T. Kurtz, Markov processes, Characterization and Convergence, Wiley, New York 1986.
  4. G. B. Folland and E. M. Stein, Hardy Spaces on Homogeneous Groups, Princeton Univ. Press, 1982.
  5. I. I. Gikhman and A. W. Skorokhod, Introduction to the Theory of Stochastic Processes, Nauka, Moscow 1965 (in Russian).
  6. P. Głowacki, A calculus of symbols and convolution semigroups on the Heisenberg group, Studia Math. 77 (1982), 291-321.
  7. P. Głowacki, Stable semigroups of measures on the Heisenberg group, ibid. 79 (1984), 105-138.
  8. P. Głowacki, Stable semi-groups of measures as commutative approximate identities on non-graded homogeneous groups, Invent. Math. 83 (1986), 557-582.
  9. A. Hulanicki, A class of convolution semi-groups of measures on a Lie group, in: Lecture Notes in Math. 828, Springer, 1980, 82-101.
  10. G. A. Hunt, Semi-groups of measures on Lie groups, Trans. Amer. Math. Soc. 81 (1956), 264-293.
  11. N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland, Amsterdam 1981.
  12. A. Janssen, Charakterisierung stetiger Faltungshalbgruppen durch das Lévy-Mass, Math. Ann. 246 (1980), 233-240.
  13. P. Malliavin, Stochastic calculus of variation and hypoelliptic operators, in: Proc. Internat. Sympos. on Stochastic Differential Equations, Kyoto 1976, K. Itô (ed.), Kinokuniya and Wiley, 1978, 195-263.
  14. P. Pazy, Semi-groups of Linear Operators and Application to Partial Differential Equations, Springer, New York 1983.
  15. D. Stroock, The Malliavin calculus and its application to second order parabolic differential equations: Part I, Math. Systems Theory 14 (1981), 25-65.
  16. F. W. Warner, Foundations of Differentiable Manifolds and Lie Groups, Springer, New York 1983.
Studia Mathematica
1991/100/3
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Pages:
183-205
Main language of publication
English
Received
1990-04-20
Published
1991
Exact and natural sciences