Abstract
Let (μ t ) t⪖0 be a τ-semistable convolution semigroup of probability measures on a Lie groupG whose idempotentμ 0 is the Haar measure on some compact subgroupK. Then all the measuresμ 1 are supported by theK-contraction groupC K(τ) of the topological automorphism τ ofG. We prove here the structure theoremC K(τ)=C(τ)K, whereC(τ) is the contraction group of τ. Then it turns out that it is sufficient to study semistable convolution semigroups on simply connected nilpotent Lie groups that have Lie algebras with a positive graduation.
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References
Bourbaki, N. (1975).Eléments de Mathématique XXXVIII. Groupes et Algèbres de Lie. Chapters 7 and 8, Actual. Scient. Ind. 1364, Hermann, Paris.
Hazod, W. (1984). Remarks on [semi-] stable probabilities. InProbability Theory on Groups VII. Proceedings, Oberwolfach 1983, pp. 182–203. Lecture Notes in Mathematics 1064, Springer, New York.
Hazod, W. (1984). Stable and semistable probabilities on groups and on vector spaces. InProbability Theory on Vector Spaces III. Proceedings, Lublin 1983, pp. 69–89. Lecture Notes in Mathematics 1080 Springer, New York.
Hazod, W., and Siebert, E. (1986). Continuous automorphism groups on a locally compact group contracting modulo a compact subgroup and applications to stable convolution semigroups. Semigroup Forum 33, pp. 111–143.
Helgason, S. (1978).Differential Geometry, Lie Groups, and Symmetric Spaces, Academic, New York.
Hewitt, E., and Ross, K. E. (1963).Abstract Harmonic Analysis I, Springer, Berlin.
Heyer, H. (1977).Probability Measures on Locally Compact Groups, Springer, Berlin.
Hochschild, G. (1965).The Structure of Lie Groups, Holden Day, San Francisco.
Jajte, R. (1977). Semi-stable probability measures on ℝN Studia Math. 61, 29–39.
Murakami, S. (1952). On the automorphisms of a real semisimple Lie algebra,J. Math. Soc. Jpn. 4, 103–133.
Parthasarathy, K. R. (1967).Probability Measures on Metric Spaces, Academic, New York.
Siebert, E. (1986). Contractive automorphisms on locally compact groups,Math. Z.,191, 73–90.
Yamabe, H. (1950). On an arcwise connected subgroup of a Lie group,Osaka Math. J. 2, 13–14.
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Hazod, W., Siebert, E. Automorphisms on a Lie group contracting modulo a compact subgroup and applications to semistable convolution semigroups. J Theor Probab 1, 211–225 (1988). https://doi.org/10.1007/BF01046936
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DOI: https://doi.org/10.1007/BF01046936