## Ergodicity for the stochastic Complex Ginzburg–Landau equations, preprint available on http://www.bretagne.ens-cachan.fr/math/people/cyril.odasso

Citations: | 10 - 6 self |

### BibTeX

@MISC{Odasso_ergodicityfor,

author = {Cyril Odasso and Ecole Normale and Supérieure De Cachan and Antenne De Bretagne},

title = {Ergodicity for the stochastic Complex Ginzburg–Landau equations, preprint available on http://www.bretagne.ens-cachan.fr/math/people/cyril.odasso},

year = {}

}

### OpenURL

### Abstract

Abstract: We study a stochastic complex Ginzburg–Landau (CGL) equation driven by a smooth noise in space and we establish exponential convergence of the Markovian transition semi-group toward a unique invariant probability measure. Since Doob Theorem does not seem not to be useful in our situation, a coupling method is used. In order to make this method easier to understand, we first focus on two simple examples which contain most of the arguments and the essential difficulties.

### Citations

303 |
Stochastic equations in infinite dimensions
- prato, G, et al.
- 1992
(Show Context)
Citation Context ...tion. We use the following assumptions. ⎧ ⎪⎨ i) There exists σ0 > 0 such that, σl(x) ≥ σ0, x ∈ R. (1.14) ⎪⎩ ii) |g(x, y1) − g(x, y2)| ≤ |y1 − y2| , (x, y1, y2) ∈ R2 . By the dissipativity method (see =-=[4]-=- section 11.5), ii) implies exponential convergence to equilibrium for the second equation if X is fixed. Whilst the coupling 6Ergodicity for the stochastic Complex Ginzburg–Landau equations argument... |

52 | Ergodicity of the 2–D Navier–Stokes equation under random perturbations - Flandoli, Maslowski - 1995 |

47 |
On the theory of superconductivity
- Ginzburg, Landau
- 1950
(Show Context)
Citation Context ... transition semigroup, invariant measure, ergodicity, coupling method, Girsanov’s formula, Foias–Prodi estimate. Introduction Originally introduced to describe a phase transition in superconductivity =-=[8]-=-, the Complex Ginzburg–Landau (CGL) equation also models the propagation of dispersive non-linear waves in various areas of physics such as hydrodynamics [18], [19], optics, plasma physics, chemical r... |

39 | Stochastic dissipative PDE’s and Gibbs measures
- Kuksin, Shirikyan
(Show Context)
Citation Context .... The unknown u is a complex valued process depending on x ∈ D, D ⊂ Rd a bounded domain, and t ≥ 0. We want to consider noises which may be degenerate and our work is in the spirit of [3], [7], [10], =-=[13]-=-, [14], [15], [16], [18] and [23]. Many ideas of this article are taken from these works. However, we develop several generalisations. The main idea is to compensate the degeneracy of the noise on som... |

38 | Gibbsian dynamics and ergodicity for the stochastically forced Navier-Stokes equation
- E, Mattingly, et al.
(Show Context)
Citation Context ..., for x ∈ D. The unknown u is a complex valued process depending on x ∈ D, D ⊂ Rd a bounded domain, and t ≥ 0. We want to consider noises which may be degenerate and our work is in the spirit of [3], =-=[7]-=-, [10], [13], [14], [15], [16], [18] and [23]. Many ideas of this article are taken from these works. However, we develop several generalisations. The main idea is to compensate the degeneracy of the ... |

37 |
Finite bandwidth, finite amplitude convection
- Newell, Whitehead
- 1969
(Show Context)
Citation Context ...be a phase transition in superconductivity [8], the Complex Ginzburg–Landau (CGL) equation also models the propagation of dispersive non-linear waves in various areas of physics such as hydrodynamics =-=[18]-=-, [19], optics, plasma physics, chemical reaction [10]... When working in non-homogenous or random media, a noise is often introduced and the stochastic CGL equation may be more representative than th... |

34 | Exponential convergence for the stochastically forced Navier-Stokes equations and other partially dissipative dynamics
- Mattingly
- 2002
(Show Context)
Citation Context ...n u is a complex valued process depending on x ∈ D, D ⊂ Rd a bounded domain, and t ≥ 0. We want to consider noises which may be degenerate and our work is in the spirit of [3], [9], [12], [13], [14], =-=[16]-=-, [17] and [20]. Many ideas of this article are taken from these works. However, we develop several generalisations. The main idea is to compensate the degeneracy of the noise on some subspaces by dis... |

26 | Exponential Mixing Properties of Stochastic PDEs Through Asymptotic
- Hairer
- 2002
(Show Context)
Citation Context ...e established in [2] and [15], respectively. The stochastic NLS equation is studied in [5] and [6]. Ergodicity of the stochastic CGL equation is established in [1] when the noise is invertible and in =-=[9]-=- for the one-dimensionnal cubic case when the noise is diagonal, does not depend on the solution and is smooth in space. Our aim in this article is to study ergodicity for stochastic CGL equation unde... |

24 | Exponential mixing for the 2D stochastic NavierStokes dynamics
- Bricmont, Kupiainen, et al.
- 2002
(Show Context)
Citation Context ...u0(x), for x ∈ D. The unknown u is a complex valued process depending on x ∈ D, D ⊂ Rd a bounded domain, and t ≥ 0. We want to consider noises which may be degenerate and our work is in the spirit of =-=[3]-=-, [9], [12], [13], [14], [16], [17] and [20]. Many ideas of this article are taken from these works. However, we develop several generalisations. The main idea is to compensate the degeneracy of the n... |

23 |
The stochastic nonlinear Schrödinger equation in H 1
- Bouard, Debussche
- 2003
(Show Context)
Citation Context ... to the NLS equation. The inviscid limits of the deterministic and stochastic CGL equation to the NLS equation are established in [2] and [15], respectively. The stochastic NLS equation is studied in =-=[5]-=- and [6]. Ergodicity of the stochastic CGL equation is established in [1] when the noise is invertible and in [9] for the one-dimensionnal cubic case when the noise is diagonal, does not depend on the... |

20 | Coupling approach to white-forced nonlinear PDEs
- Kuksin, Shirikyan
- 2002
(Show Context)
Citation Context ...unknown u is a complex valued process depending on x ∈ D, D ⊂ Rd a bounded domain, and t ≥ 0. We want to consider noises which may be degenerate and our work is in the spirit of [3], [9], [12], [13], =-=[14]-=-, [16], [17] and [20]. Many ideas of this article are taken from these works. However, we develop several generalisations. The main idea is to compensate the degeneracy of the noise on some subspaces ... |

13 |
Universal Decay of vortex density in two dimensions, Physica A 195
- Huber, Alstrom
- 1993
(Show Context)
Citation Context ...mplex Ginzburg–Landau (CGL) equation also models the propagation of dispersive non-linear waves in various areas of physics such as hydrodynamics [18], [19], optics, plasma physics, chemical reaction =-=[10]-=-... When working in non-homogenous or random media, a noise is often introduced and the stochastic CGL equation may be more representative than the deterministic one. The CGL equation arises in the sa... |

11 | On recent progress for the stochastic Navier Stokes equations. Journées “Équations aux Dérivées Partielles”, Exp - Mattingly - 2003 |

9 |
Exponential mixing for 2D Navier-Stokes equation pertubed by an unbounded noise
- Shirikyan
- 2004
(Show Context)
Citation Context ...x valued process depending on x ∈ D, D ⊂ Rd a bounded domain, and t ≥ 0. We want to consider noises which may be degenerate and our work is in the spirit of [3], [9], [12], [13], [14], [16], [17] and =-=[20]-=-. Many ideas of this article are taken from these works. However, we develop several generalisations. The main idea is to compensate the degeneracy of the noise on some subspaces by dissipativity argu... |

5 |
A coupling approach to randomly forced randomly forced PDE’s
- Kuksin, Piatnitski, et al.
- 2002
(Show Context)
Citation Context ... x ∈ D. The unknown u is a complex valued process depending on x ∈ D, D ⊂ Rd a bounded domain, and t ≥ 0. We want to consider noises which may be degenerate and our work is in the spirit of [3], [9], =-=[12]-=-, [13], [14], [16], [17] and [20]. Many ideas of this article are taken from these works. However, we develop several generalisations. The main idea is to compensate the degeneracy of the noise on som... |

4 |
Review of the finite bandwidth concept
- Newell, Whitehead
- 1971
(Show Context)
Citation Context ...hase transition in superconductivity [8], the Complex Ginzburg–Landau (CGL) equation also models the propagation of dispersive non-linear waves in various areas of physics such as hydrodynamics [18], =-=[19]-=-, optics, plasma physics, chemical reaction [10]... When working in non-homogenous or random media, a noise is often introduced and the stochastic CGL equation may be more representative than the dete... |

3 |
Inviscid limits of the
- Bebouche, Jüngel
- 2000
(Show Context)
Citation Context ...ion. In fact, the CGL equation is obtained by adding two viscous terms to the NLS equation. The inviscid limits of the deterministic and stochastic CGL equation to the NLS equation are established in =-=[2]-=- and [15], respectively. The stochastic NLS equation is studied in [5] and [6]. Ergodicity of the stochastic CGL equation is established in [1] when the noise is invertible and in [9] for the one-dime... |

2 |
Etudes théoriques et numériques sur les équations de Schrödinger et Ginzburg-Landau stochastiques , Thèse soutenue à Orsay
- Barton-Smith
- 2002
(Show Context)
Citation Context ...stic CGL equation to the NLS equation are established in [2] and [15], respectively. The stochastic NLS equation is studied in [5] and [6]. Ergodicity of the stochastic CGL equation is established in =-=[1]-=- when the noise is invertible and in [9] for the one-dimensionnal cubic case when the noise is diagonal, does not depend on the solution and is smooth in space. Our aim in this article is to study erg... |

2 | Randomly forced CGL equation: Stationnary measure and the inviscid limit - Kuksin, Shirikyan |

2 |
Invariant measure for the stochastic Ginzburg Landau equation
- Barton-Smith
(Show Context)
Citation Context ...quations equation to the NLS equation are established in [2] and [17], respectively. The stochastic NLS equation is studied in [5] and [6]. Ergodicity of the stochastic CGL equation is established in =-=[1]-=- when the noise is invertible and in [10] for the one-dimensionnal cubic case when the noise is diagonal, does not depend on the solution and is smooth in space. Our aim in this article is to study er... |

1 |
On exponential convergence to a stationnary mesure for nonlinear PDEs
- Kuksin
- 2002
(Show Context)
Citation Context ...rojector on the first N modes. The main assumption of the papers cited above as well as in this work is that the noise is non-degenerate on the space spanned by (ek)1≤k≤N for N sufficiently large. In =-=[11]-=-, [12], [13], [14] and [20], the noise is also additive, i.e. b(u) does not depend on u. The method developped in [16] and [17] allows to treat more general noises and, in [16], b is allowed to depend... |

1 |
Propriétés ergodiques de l’équation de Ginzburg-Landau complexe bruitée, Mémoire de DEA
- Odasso
- 2003
(Show Context)
Citation Context ... such that N ≥ N0” and (2.19) Eu1(t) ≤ ρ + Bt imply (2.20) |r(t)| L 2 ≤ |r(0)| L 2 exp(−2t + c1ρ), where c1 is the constant of Proposition 2.5. For the first case, this result is Proposition 1.1.6 of =-=[22]-=-. For the second case the proof is the same. Sketch of the proof of Lemma 2.16. The proof of Lemma 2.16 is similar to the proof of Proposition 2.5. Indeed it is sufficient to prove (2.21) I ′ = −((η +... |