On sequentially weakly Feller solutions to SPDE’s

Bohdan Maslowski; Jan Seidler

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1999)

  • Volume: 10, Issue: 2, page 69-78
  • ISSN: 1120-6330

Abstract

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A rather general class of stochastic evolution equations in Hilbert spaces whose transition semigroups are Feller with respect to the weak topology is found, and consequences for existence of invariant measures are discussed.

How to cite

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Maslowski, Bohdan, and Seidler, Jan. "On sequentially weakly Feller solutions to SPDE’s." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 10.2 (1999): 69-78. <http://eudml.org/doc/252296>.

@article{Maslowski1999,
abstract = {A rather general class of stochastic evolution equations in Hilbert spaces whose transition semigroups are Feller with respect to the weak topology is found, and consequences for existence of invariant measures are discussed.},
author = {Maslowski, Bohdan, Seidler, Jan},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Stochastic partial differential equations; Weakly Feller processes; Invariant measures; stochastic partial differential equations; weakly Feller processes; invariant measures},
language = {eng},
month = {6},
number = {2},
pages = {69-78},
publisher = {Accademia Nazionale dei Lincei},
title = {On sequentially weakly Feller solutions to SPDE’s},
url = {http://eudml.org/doc/252296},
volume = {10},
year = {1999},
}

TY - JOUR
AU - Maslowski, Bohdan
AU - Seidler, Jan
TI - On sequentially weakly Feller solutions to SPDE’s
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1999/6//
PB - Accademia Nazionale dei Lincei
VL - 10
IS - 2
SP - 69
EP - 78
AB - A rather general class of stochastic evolution equations in Hilbert spaces whose transition semigroups are Feller with respect to the weak topology is found, and consequences for existence of invariant measures are discussed.
LA - eng
KW - Stochastic partial differential equations; Weakly Feller processes; Invariant measures; stochastic partial differential equations; weakly Feller processes; invariant measures
UR - http://eudml.org/doc/252296
ER -

References

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  1. Brzézniak, Z. - Gątarek, D., Martingale solutions and invariant measures for stochastic evolution equations in Banach spaces. The University of Hull, Mathematics Research Reports, 11, 1998, no. 2. Zbl0996.60074
  2. Curtain, R. F. - Pritchard, A. J., Infinite dimensional linear systems theory. Lecture Notes in Control and Inform. Sci., 8, Springer-Verlag, Berlin1978. Zbl0389.93001MR516812
  3. Da Prato, G. - Gątarek, D. - Zabczyk, J., Invariant measures for semilinear stochastic equations. Stochastic Anal. Appl., 10, 1992, 387-408. Zbl0758.60057MR1178482DOI10.1080/07362999208809278
  4. Da Prato, G. - Zabczyk, J., Stochastic equations in infinite dimensions. Cambridge University Press, Cambridge1992. Zbl1140.60034MR1207136DOI10.1017/CBO9780511666223
  5. Da Prato, G. - Zabczyk, J., Ergodicity for infinite dimensional systems. Cambridge University Press, Cambridge1996. Zbl0849.60052MR1417491DOI10.1017/CBO9780511662829
  6. Ichikawa, A., Dynamic programming approach to stochastic evolution equations. SIAM J. Control Optim., 17, 1979, 152-174. Zbl0434.93069MR516862DOI10.1137/0317012
  7. Ichikawa, A., Semilinear stochastic evolution equations: Boundedness, stability and invariant measures. Stochastics, 12, 1984, 1-39. Zbl0538.60068MR738933DOI10.1080/17442508408833293
  8. LeCam, L., Convergence in distribution of stochastic processes. Univ. Calif. Publ. Statist., 2, 1957, 207-236. Zbl0077.12301MR86117
  9. Leha, G. - Ritter, G., Lyapunov-type conditions for stationary distributions of diffusion processes on Hilbert spaces. Stochastics Stochastics Rep., 48, 1994, 195-225. Zbl0828.60063MR1782748
  10. Lucchetti, R. - Patrone, F., On Nemytskii’s operator and its application to the lower semicontinuity of integral functionals. Indiana Univ. Math. J., 29, 1980, 703-713. Zbl0476.47049MR589437DOI10.1512/iumj.1980.29.29051
  11. Maslowski, B. - Seidler, J., Probabilistic approach to the strong Feller property. To appear. Zbl0966.60076MR1790081DOI10.1007/s440-000-8014-0
  12. Seidler, J. - Vrkoč, I., An averaging principle for stochastic evolution equations. Časopis Pěst. Mat., 115, 1990, 240-263. Zbl0718.60068MR1071056
  13. Vrkoč, I., A dynamical system in a Hilbert space with a weakly attractive nonstationary point. Math. Bohem., 118, 1993, 401-423. Zbl0794.34054MR1251884
  14. Zabczyk, J., On optimal stochastic control of discrete-time systems in Hilbert space. SIAM J. Control, 13, 1975, 1217-1234. Zbl0313.93067MR384291

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