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Title

On a class of Markov type semigroups in spaces of uniformly continuous and bounded functions

Authors 1

Affiliations

  1. Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy

Abstract

We study a new class of Markov type semigroups (not strongly continuous in general) in the space of all real, uniformly continuous and bounded functions on a separable metric space E. Our results allow us to characterize the generators of Markov transition semigroups in infinite dimensions such as the heat and the Ornstein-Uhlenbeck semigroups.

Bibliography

  1. W. Arendt, Vector valued Laplace transforms and Cauchy problems, Israel J. Math. 59 (1987), 327-352.
  2. R. B. Ash, Real Analysis and Probability, Academic Press, New York, 1972.
  3. P. Cannarsa and G. Da Prato, Infinite dimensional elliptic equations with Hölder continuous coefficients, Adv. Differential Equations 1 (1996), 425-452.
  4. P. Cannarsa and G. Da Prato, Potential theory in Hilbert spaces, in: Proc. Sympos. Appl. Math. 54, Amer. Math. Soc., 1998, 27-51.
  5. S. Cerrai, A Hille-Yosida theorem for weakly continuous semigroups, Semigroup Forum 49 (1994), 349-367.
  6. S. Cerrai and F. Gozzi, Strong solutions of Cauchy problems associated to weakly continuous semigroups, Differential Integral Equations 8 (1994), 465-486.
  7. G. Da Prato and A. Lunardi, On the Ornstein-Uhlenbeck operator in spaces of continuous functions, J. Funct. Anal. 131 (1995), 94-114.
  8. G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia Math. Appl. 44, Cambridge Univ. Press, 1992.
  9. J. Diestel, Sequences and Series in Banach Spaces, Grad. Text. in Math. 92, Springer, New York, 1984.
  10. E. B. Dynkin, Markov Processes, Vol. I, Springer, Berlin, 1965.
  11. N. Ethier and T. G. Kurtz, Markov Processes, Characterization and Convergence, Wiley, 1986.
  12. M. Fuhrman and M. Röckner, Generalized Mehler semigroups: the non-Gaussian case, Potential Anal., to appear.
  13. L. Gross, Potential theory on Hilbert space, J. Funct. Anal. 1 (1967), 123-181.
  14. P. Guiotto, Non-differentiability of heat semigroups in infinite dimensional Hilbert spaces, Semigroup Forum 55 (1997), 232-236.
  15. M. Hieber and H. Kellerman, Integrated semigroups, J. Funct. Anal. 84 (1989), 160-180.
  16. B. Jefferies, Weakly integrable semigroups on locally convex spaces, ibid. 66 (1986), 347-364.
  17. B. Jefferies, The generation of weakly integrable semigroups, ibid. 73 (1987), 195-215.
  18. H. H. Kuo, Gaussian Measures in Banach Spaces, Lecture Notes in Math. 463, Springer, 1975.
  19. A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, 1995.
  20. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983.
  21. E. Priola, Schauder estimates for a homogeneous Dirichlet problem in a half space of a Hilbert space, Nonlinear Anal., to appear.
  22. E. Priola, π-Semigroups and applications, preprint n. 9, Scuola Norm. Sup. Pisa, 1998.
  23. K. Yosida, Functional Analysis, 4th ed., Springer, Berlin, 1974.
  24. L. Zambotti, A new approach to existence and uniqueness for martingale problems in infinite dimensions, preprint n. 13, Scuola Norm. Sup. Pisa, 1998.
Studia Mathematica
1999/136/3
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Pages:
271-295
Main language of publication
English
Received
1998-07-17
Accepted
1999-02-15
Published
1999
Exact and natural sciences