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We study a damped stochastic non-linear Schr\"{o}dinger (NLS) equation driven by an additive noise. It is white in time and smooth in space. Using a coupling method, we establish convergence of the Markovian transition semi-group toward a…
We study the long-time mixing behavior of the stochastic nonlinear Schr\"odinger equation in $\mathbb{R}^d$, $d\le 3$. It is well known that, under a sufficiently strong damping force, the system admits unique ergodicity, although the rate…
We prove the exponential convergence to a unique invariant measure for locally damped nonlinear Schr\"odinger equations, perturbed by bounded noise acting on only two Fourier modes. To tackle the lack of smoothing effect, we introduce…
This paper investigates exponential mixing of the invariant measure for randomly forced nonlinear Schr\"{o}dinger equation, with damping and random noise localized in space. Our study emphasizes the crucial role of exponential asymptotic…
We establish a general criterion which ensures exponential mixing of parabolic Stochastic Partial Differential Equations (SPDE) driven by a non additive noise which is white in time and smooth in space. We apply this criterion on two…
This paper studies the 1D stochastic Allen--Cahn equation on a bounded domain driven by localized white noise. We prove that the associated Markov process admits a unique invariant measure and is exponential mixing. The main challenge lies…
Our goal in this paper is to investigate ergodicity of the white-forced wave equation on the whole line. Under the assumption that sufficiently many directions of the phase space are stochastically forced, we prove the uniqueness of…
We study the long time statistics of a class of semi--linear wave equations modeling the motions of a particle suspended in continuous media while being subjected to random perturbations via an additive Gaussian noise. By comparison with…
We consider a stochastic nonlinear defocusing Schr\"{o}dinger equation with zero-order linear damping, where the stochastic forcing term is given by a combination of a linear multiplicative noise in the Stratonovich form and a nonlinear…
In this article, we have analyzed the full discretization of the Stochastic semilinear Schr\"{o}dinger equation in a bounded convex polygonal domain driven by multiplicative Wiener noise. We use the finite element method for spatial…
A standard finite element method discretizes the stochastic linear Schr\"{o}dinger equation driven by additive noise in the spatial variables. The weak convergence of the resulting approximate solution is analyzed, and it is established…
We study the ergodicity of finite-dimensional approximations of the Schr\"odinger equation. The system is driven by a multiplicative scalar noise. Under general assumptions over the distribution of the noise, we show that the system has a…
We study the nonlinear Schr\"odinger equation with linear damping, i.e. a zero order dissipation, and additive noise. Working in $R^d$ with d = 2 or d = 3, we prove the uniqueness of the invariant measure when the damping coefficient is…
In this paper, we investigate the stochastic damped Burgers equation with multiplicative space-time white noise defined on the entire real line. We prove the existence and uniqueness of a mild solution of the stochastic damped Burgers…
We consider parabolic stochastic partial differential equations driven by white noise in time. We prove exponential convergence of the transition probabilities towards a unique invariant measure under suitable conditions. These conditions…
We study the Navier-Stokes equations in dimension 3 (NS3D) driven by a noise which is white in time. We establish that if the noise is at same time sufficiently smooth and non degenerate in space, then the weak solutions converge…
We study the stochastic nonlinear Schr\"odinger equations with additive stochastic forcing. By using the dispersive estimate, we present a simple argument, constructing a unique local-in-time solution with rougher stochastic forcing than…
In this paper, we establish the existence and uniqueness of invariant measures for a class of semilinear stochastic partial differential equations driven by multiplicative noise on a bounded domain. The main results can be applied to SPDEs…
We proved that there exists a unique invariant measure for solutions of stochastic conservation laws with Dirichlet boundary condition driven by multiplicative noise. Moreover, a polynomial mixing property is established. This is done in…
We study the stochastic 3D primitive equations of the atmospheric mechanics. We consider them under a bounded and non-degenerate noise, which is statistically periodic in time with period $1$. In such a case we prove that the associated…