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1991 | 100 | 3 | 183-205
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Malliavin calculus for stable processes on homogeneous groups

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Let ${μ_t}_{t>0}$ 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 $μ_t$ have smooth densities.
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Twórcy
  • Institute of Mathematics, Wrocław Technical University, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland
Bibliografia
  • [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.
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