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

Abstract

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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.

How to cite

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Bergé, 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

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