# On the long-time behaviour of a class of parabolic SPDE’s : monotonicity methods and exchange of stability

Benjamin Bergé; Bruno Saussereau

ESAIM: Probability and Statistics (2005)

- Volume: 9, page 254-276
- ISSN: 1292-8100

## Access Full Article

top## Abstract

top## How to cite

topBergé, Benjamin, and Saussereau, Bruno. "On the long-time behaviour of a class of parabolic SPDE’s : monotonicity methods and exchange of stability." ESAIM: Probability and Statistics 9 (2005): 254-276. <http://eudml.org/doc/245515>.

@article{Bergé2005,

abstract = {In this article we prove new results concerning the structure and the stability properties of the global attractor associated with a class of nonlinear parabolic stochastic partial differential equations driven by a standard multidimensional brownian motion. We first use monotonicity methods to prove that the random fields either stabilize exponentially rapidly with probability one around one of the two equilibrium states, or that they set out to oscillate between them. In the first case we can also compute exactly the corresponding Lyapunov exponents. The last case of our analysis reveals a phenomenon of exchange of stability between the two components of the global attractor. In order to prove this asymptotic property, we show an exponential decay estimate between the random field and its spatial average under an additional uniform ellipticity hypothesis.},

author = {Bergé, Benjamin, Saussereau, Bruno},

journal = {ESAIM: Probability and Statistics},

keywords = {parabolic stochastic partial differential equations; asymptotic behaviour; monotonicity methods; Parabolic stochastic partial differential equations},

language = {eng},

pages = {254-276},

publisher = {EDP-Sciences},

title = {On the long-time behaviour of a class of parabolic SPDE’s : monotonicity methods and exchange of stability},

url = {http://eudml.org/doc/245515},

volume = {9},

year = {2005},

}

TY - JOUR

AU - Bergé, Benjamin

AU - Saussereau, Bruno

TI - On the long-time behaviour of a class of parabolic SPDE’s : monotonicity methods and exchange of stability

JO - ESAIM: Probability and Statistics

PY - 2005

PB - EDP-Sciences

VL - 9

SP - 254

EP - 276

AB - In this article we prove new results concerning the structure and the stability properties of the global attractor associated with a class of nonlinear parabolic stochastic partial differential equations driven by a standard multidimensional brownian motion. We first use monotonicity methods to prove that the random fields either stabilize exponentially rapidly with probability one around one of the two equilibrium states, or that they set out to oscillate between them. In the first case we can also compute exactly the corresponding Lyapunov exponents. The last case of our analysis reveals a phenomenon of exchange of stability between the two components of the global attractor. In order to prove this asymptotic property, we show an exponential decay estimate between the random field and its spatial average under an additional uniform ellipticity hypothesis.

LA - eng

KW - parabolic stochastic partial differential equations; asymptotic behaviour; monotonicity methods; Parabolic stochastic partial differential equations

UR - http://eudml.org/doc/245515

ER -

## References

top- [1] L. Arnold, Stochastic Differential Equations: Theory and Applications. John Wiley and Sons, New York (1974). Zbl0278.60039MR443083
- [2] D.G. Aronson and H.F. Weinberger, Nonlinear dynamics in population genetics, combustion and nerve pulse propagation. Lect. Notes Math. 446 (1975) 5–49. Zbl0325.35050
- [3] B. Bergé, I.D. Chueshov and P.A. Vuillermot, On the behavior of solutions to certain parabolic SPDE’s driven by Wiener processes. Stoch. Proc. Appl. 92 (2001) 237–263. Zbl1047.60062
- [4] H. Brézis, Analyse fonctionnelle, théorie et applications. Masson, Paris (1993). Zbl0511.46001MR697382
- [5] I.D. Chueshov, Monotone Random Systems: Theory and Applications. Lect. Notes Math., Springer, Berlin 1779 (2002). Zbl1023.37030MR1902500
- [6] I.D. Chueshov and P.A. Vuillermot, 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. Zbl0914.35021
- [7] I.D. Chueshov and P.A. Vuillermot, 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. Zbl0968.60058
- [8] I.I. Gihman and A.V. Skorohod, Stochastic Differential Equations. Ergebnisse der Mathematik und ihrer Grenzgebiete, Bd. 72. Springer, Berlin (1972). Zbl0242.60003MR346904
- [9] G. Hetzer, W. Shen and S. Zhu, Asymptotic behavior of positive solutions of random and stochastic parabolic equations of fisher and Kolmogorov type. J. Dyn. Diff. Eqs. 14 (2002) 139–188. Zbl0994.35125
- [10] R.Z. Hasminskii, Stochastic Stability of Differentiel Equations. Alphen, Sijthoff and Nordhof (1980). MR600653
- [11] N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland Mathematical Library. North-Holland, Kodansha 24 (1981). Zbl0495.60005MR1011252
- [12] 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. Zbl0018.32106
- [13] R. Manthey and K. Mittmann, On a class of stochastic functionnal-differential equations arising in population dynamics. Stoc. Stoc. Rep. 64 (1998) 75–115. Zbl0907.60059
- [14] J.D. Murray, Mathematical Biology. Second Edition. Springer, Berlin 19 (1993). Zbl0779.92001MR1239892
- [15] B. Øksendal, G. Våge and H.Z. Zhao, Asymptotic properties of the solutions to stochastic KPP equations. Proc. Roy. Soc. Edinburgh 130A (2000) 1363–1381. Zbl0978.60070
- [16] B. Øksendal, G. Våge and H.Z. Zhao, Two properties of stochastic KPP equations: ergodicity and pathwise property. Nonlinearity 14 (2001) 639–662. Zbl0993.60064
- [17] M. Sanz-Solé and P.A. Vuillermot, 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. Zbl1026.60080

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.