Gibbsian dynamics and invariant measures for stochastic dissipative PDEs

E. Weinan; Di Liu

doi:10.1023/A:1019747716056

Gibbsian dynamics and invariant measures for stochastic dissipative PDEs

E. Weinan, Di Liu

Mathematics

Research output: Contribution to journal › Article › peer-review

21 Scopus citations

Abstract

We present a general strategy for proving ergodicity for stochastically forced nonlinear dissipative PDEs. It consists of two main steps. The first step is the reduction to a finite dimensional Gibbsian dynamics of the low modes. The second step is to prove the equivalence between measures induced by different past histories using Girsanov theorem. As applications, we prove ergodicity for Ginzburg-Landau, Kuramoto-Sivashinsky and Cahn-Hilliard equations with stochastic forcing.

Original language	English (US)
Pages (from-to)	1125-1156
Number of pages	32
Journal	Journal of Statistical Physics
Volume	108
Issue number	5-6
DOIs	https://doi.org/10.1023/A:1019747716056
State	Published - 2002

All Science Journal Classification (ASJC) codes

Statistical and Nonlinear Physics
Mathematical Physics

Keywords

Ergodicity
Infinite-dimensional random dynamical systems
Invariant measures
Stationary processes
Stochastic partial differential equations

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title = "Gibbsian dynamics and invariant measures for stochastic dissipative PDEs",

abstract = "We present a general strategy for proving ergodicity for stochastically forced nonlinear dissipative PDEs. It consists of two main steps. The first step is the reduction to a finite dimensional Gibbsian dynamics of the low modes. The second step is to prove the equivalence between measures induced by different past histories using Girsanov theorem. As applications, we prove ergodicity for Ginzburg-Landau, Kuramoto-Sivashinsky and Cahn-Hilliard equations with stochastic forcing.",

keywords = "Ergodicity, Infinite-dimensional random dynamical systems, Invariant measures, Stationary processes, Stochastic partial differential equations",

author = "E. Weinan and Di Liu",

note = "Funding Information: We are very grateful to J. Mattingly and Ya. Sinai for many stimulating discussions concerning the topics here. Weinan E{\textquoteright}s work is partially supported by NSF via a PFF award.",

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doi = "10.1023/A:1019747716056",

language = "English (US)",

volume = "108",

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journal = "Journal of Statistical Physics",

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T1 - Gibbsian dynamics and invariant measures for stochastic dissipative PDEs

AU - Weinan, E.

AU - Liu, Di

N1 - Funding Information: We are very grateful to J. Mattingly and Ya. Sinai for many stimulating discussions concerning the topics here. Weinan E’s work is partially supported by NSF via a PFF award.

PY - 2002

Y1 - 2002

N2 - We present a general strategy for proving ergodicity for stochastically forced nonlinear dissipative PDEs. It consists of two main steps. The first step is the reduction to a finite dimensional Gibbsian dynamics of the low modes. The second step is to prove the equivalence between measures induced by different past histories using Girsanov theorem. As applications, we prove ergodicity for Ginzburg-Landau, Kuramoto-Sivashinsky and Cahn-Hilliard equations with stochastic forcing.

AB - We present a general strategy for proving ergodicity for stochastically forced nonlinear dissipative PDEs. It consists of two main steps. The first step is the reduction to a finite dimensional Gibbsian dynamics of the low modes. The second step is to prove the equivalence between measures induced by different past histories using Girsanov theorem. As applications, we prove ergodicity for Ginzburg-Landau, Kuramoto-Sivashinsky and Cahn-Hilliard equations with stochastic forcing.

KW - Ergodicity

KW - Infinite-dimensional random dynamical systems

KW - Invariant measures

KW - Stationary processes

KW - Stochastic partial differential equations

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SN - 0022-4715

VL - 108

SP - 1125

EP - 1156

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

IS - 5-6

ER -

Gibbsian dynamics and invariant measures for stochastic dissipative PDEs

Abstract

All Science Journal Classification (ASJC) codes

Keywords

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