# 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 (2010)

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

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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 (2010): 254-276. <http://eudml.org/doc/104336>.

@article{Bergé2010,

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.; asymptotic behaviour; monotonicity methods},

language = {eng},

month = {3},

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/104336},

volume = {9},

year = {2010},

}

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

DA - 2010/3//

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.; asymptotic behaviour; monotonicity methods

UR - http://eudml.org/doc/104336

ER -

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