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On the long-time behaviour of a class of parabolic SPDE's: monotonicity methodsand exchange of stability

Published online by Cambridge University Press:  15 November 2005

Benjamin Bergé  and
Bruno Saussereau
Benjamin Bergé
Affiliation:
Institut de Mathématiques, Université de Neuchâtel, Rue Émile Argand, 11, 2007 Neuchâtel, Switzerland; [email protected]
Bruno Saussereau
Affiliation:
Département de Mathématiques, Université de Franche-Comté, 16 route de Gray, 25030 Besançon Cedex, France; [email protected]
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Abstract

In this article we prove new results concerning thestructure and the stability properties of the global attractor associatedwith a class of nonlinear parabolic stochastic partial differential equationsdriven by a standard multidimensional Brownian motion.We first use monotonicity methodsto prove that the random fields either stabilize exponentially rapidly withprobability one around one of the two equilibrium states, or that they set outto oscillate between them. In the first case we can also compute exactly thecorresponding Lyapunov exponents.The last case of our analysis reveals a phenomenon of exchange of stabilitybetween the two components of the global attractor. In order to prove thisasymptotic property, we show an exponential decay estimate between the randomfield and its spatial average under an additional uniform ellipticityhypothesis.

Keywords

Parabolic stochastic partial differential equationsasymptotic behaviourmonotonicity methods.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 9 , February 2005 , pp. 254 - 276
Copyright
© EDP Sciences, SMAI, 2005

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References

L. Arnold, Stochastic Differential Equations: Theory and Applications. John Wiley and Sons, New York (1974).
Aronson, D.G. and Weinberger, H.F., Nonlinear dynamics in population genetics, combustion and nerve pulse propagation. Lect. Notes Math. 446 (1975) 5–49. CrossRef
Bergé, B., Chueshov, I.D. and Vuillermot, P.A., On the behavior of solutions to certain parabolic SPDE's driven by Wiener processes. Stoch. Proc. Appl. 92 (2001) 237–263. CrossRef
H. Brézis, Analyse fonctionnelle, théorie et applications. Masson, Paris (1993).
I.D. Chueshov, Monotone Random Systems: Theory and Applications. Lect. Notes Math., Springer, Berlin 1779 (2002). CrossRef
Chueshov, I.D. and Vuillermot, P.A., Long-time behavior of solutions to a class of stochastic parabolic equations with homogeneous white noise: Stratonovitch's case. Probab. Theory Relat. Fields 112 (1998) 149–202. CrossRef
Chueshov, I.D. and Vuillermot, P.A., Long-time behavior of solutions to a class of stochastic parabolic equations with homogeneous white noise: Itô's case. Stochastic Anal. Appl. 18 (2000) 581–615. CrossRef
I.I. Gihman and A.V. Skorohod, Stochastic Differential Equations. Ergebnisse der Mathematik und ihrer Grenzgebiete, Bd. 72. Springer, Berlin (1972).
Hetzer, G., Shen, W. and Zhu, S., Asymptotic behavior of positive solutions of random and stochastic parabolic equations of fisher and Kolmogorov type. J. Dyn. Diff. Eqs. 14 (2002) 139–188. CrossRef
R.Z. Hasminskii, Stochastic Stability of Differentiel Equations. Alphen, Sijthoff and Nordhof (1980).
N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland Mathematical Library. North-Holland, Kodansha 24 (1981).
A. Kolmogoroff, I. Petrovsky and N. Piscounoff, Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique. Bull. de l'Univ. d'État à Moscou, série internationale 1 (1937) 1–25.
Manthey, R. and Mittmann, K., On a class of stochastic functionnal-differential equations arising in population dynamics. Stoc. Stoc. Rep. 64 (1998) 75–115. CrossRef
J.D. Murray, Mathematical Biology. Second Edition. Springer, Berlin 19 (1993).
Øksendal, B., Våge, G. and Zhao, H.Z., Asymptotic properties of the solutions to stochastic KPP equations. Proc. Roy. Soc. Edinburgh 130A (2000) 1363–1381. CrossRef
Øksendal, B., Våge, G. and Zhao, H.Z., Two properties of stochastic KPP equations: ergodicity and pathwise property. Nonlinearity 14 (2001) 639–662. CrossRef
Sanz-Solé, M. and Vuillermot, P.A., Equivalence and Hölder-Sobolev regularity of solutions for a class of non-autonomous stochastic partial differential equations. Ann. Inst. H. Poincaré Probab. Statist. 39 (2003) 703–742. CrossRef