Download PDF - Analyticity of transition semigroups and closability of bilinear forms in Hilbert spacesAll rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

Analyticity of transition semigroups and closability of bilinear forms in Hilbert spaces

Authors 1

Affiliations

  1. Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci, 32 20133 Milano, Italy

Abstract

We consider a semigroup acting on real-valued functions defined in a Hilbert space H, arising as a transition semigroup of a given stochastic process in H. We find sufficient conditions for analyticity of the semigroup in the L2(μ) space, where μ is a gaussian measure in H, intrinsically related to the process. We show that the infinitesimal generator of the semigroup is associated with a bilinear closed coercive form in L2(μ). A closability criterion for such forms is presented. Examples are also given.

Bibliography

  1. J. Bergh and J. Löfström, Interpolation Spaces, Springer, 1976.
  2. S. Cerrai, A Hille-Yosida theorem for weakly continuous semigroups, preprint, Scuola Normale Superiore di Pisa, 1993.
  3. S. Cerrai and F. Gozzi, Strong solutions of Cauchy problems associated to weakly continuous semigroups, preprint, Scuola Normale Superiore di Pisa, 1993.
  4. G. Da Prato and P. Grisvard, Sommes d'opérateurs linéaires et équations différentielles opérationnelles, J. Math. Pures Appl. 54 (1975), 305-387.
  5. G. Da Prato and J. Zabczyk, Regular densities of invariant measures in Hilbert spaces, preprint, Scuola Normale Superiore di Pisa, 1993.
  6. G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia Math. Appl. 44, Cambridge University Press, 1992.
  7. M. Fuhrman, Analyticity of transition semigroups and closability of bilinear forms in Hilbert spaces, preprint, Dipartimento di Matematica, Politecnico di Milano, n. 105/p, ottobre 1993.
  8. Z. M. Ma and M. Röckner, Dirichlet Forms, Springer, 1992.
  9. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, 1983.
  10. B. Schmuland, Non-symmetric Ornstein-Uhlenbeck processes in Banach space via Dirichlet forms, Canad. J. Math. 45 (1993), 1324-1338.
  11. A. Yagi, Coïncidence entre des espaces d'interpolation et des domaines de puissances fractionnaires d'opérateurs, C. R. Acad. Sci. Paris Sér. I Math. 299 (1984), 173-176.
  12. J. Zabczyk, Symmetric solutions of semilinear stochastic equations, in: Stochastic Partial Differential Equations and Applications, G. Da Prato and L. Tubaro (eds.), Lecture Notes in Math. 1390, Springer, 1989, 237-256.
Studia Mathematica
1995/115/1
Export to PBN, Opens in new tab
Pages:
53-71
Main language of publication
English
Received
1994-03-02
Accepted
1994-09-13
Published
1995
Exact and natural sciences