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

Marco Fuhrman

Studia Mathematica (1995)

  • Volume: 115, Issue: 1, page 53-71
  • ISSN: 0039-3223

Abstract

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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 L 2 ( 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 L 2 ( μ ) . A closability criterion for such forms is presented. Examples are also given.

How to cite

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Fuhrman, Marco. "Analyticity of transition semigroups and closability of bilinear forms in Hilbert spaces." Studia Mathematica 115.1 (1995): 53-71. <http://eudml.org/doc/216198>.

@article{Fuhrman1995,
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 $L^2(μ)$ 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 $L^2(μ)$. A closability criterion for such forms is presented. Examples are also given.},
author = {Fuhrman, Marco},
journal = {Studia Mathematica},
keywords = {semigroup acting on real-valued functions defined in a Hilbert space; transition semigroup; stochastic process; Gaussian measure; bilinear closed coercive form; closability criterion},
language = {eng},
number = {1},
pages = {53-71},
title = {Analyticity of transition semigroups and closability of bilinear forms in Hilbert spaces},
url = {http://eudml.org/doc/216198},
volume = {115},
year = {1995},
}

TY - JOUR
AU - Fuhrman, Marco
TI - Analyticity of transition semigroups and closability of bilinear forms in Hilbert spaces
JO - Studia Mathematica
PY - 1995
VL - 115
IS - 1
SP - 53
EP - 71
AB - 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 $L^2(μ)$ 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 $L^2(μ)$. A closability criterion for such forms is presented. Examples are also given.
LA - eng
KW - semigroup acting on real-valued functions defined in a Hilbert space; transition semigroup; stochastic process; Gaussian measure; bilinear closed coercive form; closability criterion
UR - http://eudml.org/doc/216198
ER -

References

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  1. [1] J. Bergh and J. Löfström, Interpolation Spaces, Springer, 1976. Zbl0344.46071
  2. [2] S. Cerrai, A Hille-Yosida theorem for weakly continuous semigroups, preprint, Scuola Normale Superiore di Pisa, 1993. 
  3. [3] S. Cerrai and F. Gozzi, Strong solutions of Cauchy problems associated to weakly continuous semigroups, preprint, Scuola Normale Superiore di Pisa, 1993. Zbl0822.47040
  4. [4] G. Da Prato and P. Grisvard, Sommes d'opérateurs linéaires et équations différentielles opérationnelles, J. Math. Pures Appl. 54 (1975), 305-387. Zbl0315.47009
  5. [5] G. Da Prato and J. Zabczyk, Regular densities of invariant measures in Hilbert spaces, preprint, Scuola Normale Superiore di Pisa, 1993. 
  6. [6] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia Math. Appl. 44, Cambridge University Press, 1992. Zbl0761.60052
  7. [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. [8] Z. M. Ma and M. Röckner, Dirichlet Forms, Springer, 1992. 
  9. [9] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, 1983. 
  10. [10] B. Schmuland, Non-symmetric Ornstein-Uhlenbeck processes in Banach space via Dirichlet forms, Canad. J. Math. 45 (1993), 1324-1338. Zbl0801.31003
  11. [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. Zbl0563.46042
  12. [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. 

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