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On the equation μ = S t μ * μ t

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Séminaire de Probabilités XLII

Part of the book series: Lecture Notes in Mathematics ((SEMPROBAB,volume 1979))

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Abstract

We discuss solutions of equation μ = S t μ*μ t and study their structure. The relationship with Ornstein-Uhlenbeck processes will also be considered.

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References

  1. Applebaum, D.: Martingale-valued measures, Ornstein-Uhlenbeck processes with jumps and operator self-decomposability in Hilbert space. In Memoriam Paul-André Meyer, Séminaire de Probabilités 39, ed. M.Emery and M.Yor, Lecture Notes in Math Vol., 1874, 173–198 Springer-Verlag (2006)

    Google Scholar 

  2. Applebaum, D.: On the infinitesimal generators of Ornstein-Uhlenbeck processes with jumps in Hilbert Space. Potential Analysis, Vol., 26, 79–100 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bogachev, V.I., Röckner, M., Schmuland, B.: Generalized Mehler semigroups and applications. Probab. Theory Relat Fields, 105, 193–225 (1996)

    Article  MATH  Google Scholar 

  4. Chojnowska-Michalik, A.: On processes of Ornstern-Uhlenbeck type in Hilbert space. Stochastics, Vol., 21, 251–286 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  5. Da Prato, G., Zabczyk, J.: Second Order partial Differential Equation in Hilbert Space. Cambridge University Press (2002)

    Google Scholar 

  6. Fuhrman, M., Röckner, M.: Generalized Mehler semigroups: the non-Gaussian case. Potential Anal., 12, 1–47 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  7. Goldys, B., Kocan, M.: Diffusion Semigroups in Spaces of Continuous Functions with Mixed Topology. Journal of Differential Equations, 173, 17–39 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  8. Heyer, H.: Structural aspects in the theory of probability: a primer in probabilities on algebraic-topological structures. World Scientific (2004)

    Google Scholar 

  9. Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes (Second Edition). Amsterdam: North-Holland Pub.Co. (1989)

    MATH  Google Scholar 

  10. Jurek, Z.J.: An integral representation of operator-selfdecomposable random variables. Bull. Acad. Pol. Sci., 30, 385–393 (1982)

    MathSciNet  MATH  Google Scholar 

  11. Jurek, Z.J., Vervaat, W.: An integral representation for selfdcomposable Banach space valued random variables. Z. Wahrscheinlichkeitstheorie verw. Gebiete, 62, 247–262 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  12. Jurek, Z.J., Mason, J.D.: Operator-Limit Distributions in Probability Theory. John Wiley and Sons.Inc (1993)

    Google Scholar 

  13. Jurek, Z.J.: Measure valued cocycles from my papers in 1982 and 1983 and Mehler semigroups. www.math.uni.wroc.pl/ zjjurek (2004)

  14. Jurek, Z.J.: Remarks on relations between Urbanik and Mehler semigroups. www.math.uni.wroc.pl/ zjjurek (2007)

  15. Lukacs, E.: Characteristic functions. Griffin's Statistical Monographs and Courses, 5. Charles Griffin, London (1960)

    Google Scholar 

  16. Parthasarathy, K.R.: Probability measures on metric spaces. Academic Press (1967)

    Google Scholar 

  17. Schmuland, B., Sun, W.: On the equation μ s + t = μ s * T s μ t . Stat. Prob. Lett., 52, 183–188 (2001)

    Article  Google Scholar 

  18. Sato, K-I., Yamazato, M.: Operator-Selfdecomposable distributions as limit distributions of processes of Ornstern-Uhlenbeck type. Stochastic Processes and Their Applications, 17, 73–100 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  19. Urbanik, K.: Lévy's probability measures on Banach spaces. Studia Math, 63, 283–308 (1978)

    MathSciNet  MATH  Google Scholar 

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Correspondence to Fangjun Xu* .

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Xu*, F. (2009). On the equation μ = S t μ * μ t . In: Donati-Martin, C., Émery, M., Rouault, A., Stricker, C. (eds) Séminaire de Probabilités XLII. Lecture Notes in Mathematics(), vol 1979. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-01763-6_4

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