Asymptotic behaviour of stochastic quasi dissipative systems
ESAIM: Control, Optimisation and Calculus of Variations, Tome 8 (2002) , pp. 587-602.

We prove uniqueness of the invariant measure and the exponential convergence to equilibrium for a stochastic dissipative system whose drift is perturbed by a bounded function.

DOI : https://doi.org/10.1051/cocv:2002038
Classification : 47D07,  35K90
Mots clés : stochastic systems, reaction-diffusion equations, invariant measures 
@article{COCV_2002__8__587_0,
     author = {Prato, Giuseppe Da},
     title = {Asymptotic behaviour of stochastic quasi dissipative systems},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {587--602},
     publisher = {EDP-Sciences},
     volume = {8},
     year = {2002},
     doi = {10.1051/cocv:2002038},
     zbl = {1064.47047},
     mrnumber = {1932964},
     language = {en},
     url = {http://www.numdam.org/item/COCV_2002__8__587_0/}
}
Prato, Giuseppe Da. Asymptotic behaviour of stochastic quasi dissipative systems. ESAIM: Control, Optimisation and Calculus of Variations, Tome 8 (2002) , pp. 587-602. doi : 10.1051/cocv:2002038. http://www.numdam.org/item/COCV_2002__8__587_0/

[1] J.M. Bismut, Large deviations and the Malliavin Calculus. Birkhäuser (1984). | MR 755001 | Zbl 0537.35003

[2] H. Brézis, Opérateurs maximaux monotones. North-Holland, Amsterdam (1973).

[3] S. Cerrai, A Hille-Yosida theorem for weakly continuous semigroups. Semigroup Forum 49 (1994) 349-367. | Zbl 0817.47048

[4] S. Cerrai, Second order PDE's in finite and infinite dimensions. A probabilistic approach. Springer, Lecture Notes in Math. 1762 (2001). | Zbl 0983.60004

[5] S. Cerrai, Optimal control problems for stochastic reaction-diffusion systems with non Lipschitz coefficients. SIAM J. Control Optim. 39 (2001) 1779-1816. | MR 1825865 | Zbl 0987.60073

[6] S. Cerrai, Stationary Hamilton-Jacobi equations in Hilbert spaces and applications to a stochastic optimal control problem. SIAM J. Control Optim. (to appear). | Zbl 0992.60066

[7] G. Da Prato, Stochastic evolution equations by semigroups methods. Centre de Recerca Matematica, Barcelona, Quaderns 11 (1998).

[8] G. Da Prato, A. Debussche and B. Goldys, Invariant measures of non symmetric dissipative stochastic systems. Probab. Theor. Related Fields (to appear). | MR 1918538

[9] G. Da Prato, D. Elworthy and J. Zabczyk, Strong Feller property for stochastic semilinear equations. Stochastic Anal. Appl. 13 (1995) 35-45. | MR 1313205 | Zbl 0817.60081

[10] G. Da Prato and M. Röckner, Singular dissipative stochastic equations in Hilbert spaces, Preprint. S.N.S. Pisa (2001). | MR 1936019 | Zbl 1036.47029

[11] G. Da Prato and J. Zabczyk, Stochastic equations in infinite dimensions. Cambridge University Press (1992). | MR 1207136 | Zbl 0761.60052

[12] G. Da Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems. Cambridge University Press, London Math. Soc. Lecture Notes 229 (1996). | MR 1417491 | Zbl 0849.60052

[13] G. Da Prato and J. Zabczyk, Differentiability of the Feynman-Kac semigroup and a control application. Rend. Mat. Accad. Lincei. 8 (1997) 183-188. | Zbl 0910.93025

[14] E.B. Dynkin, Markov Processes, Vol. I. Springer-Verlag (1965). | Zbl 0132.37901

[15] K.D. Elworthy, Stochastic flows on Riemannian manifolds, edited by M.A. Pinsky and V. Wihstutz. Birkhäuser, Diffusion Processes and Related Problems in Analysis II (1992) 33-72. | MR 1187985 | Zbl 0758.58035

[16] W.H. Fleming and H.M. Soner, Controlled Markov processes and viscosity solutions. Springer-Verlag (1993). | MR 1199811 | Zbl 0773.60070

[17] T. Kato, Nonlinear semigroups and evolution equations. J. Math. Soc. Japan 10 (1967) 508-520. | MR 226230 | Zbl 0163.38303

[18] K.R. Parthasarathy, Probability measures on metric spaces. Academic Press (1967). | MR 226684 | Zbl 0153.19101