Arnaud Debussche

Ergodicity for the 3D stochastic Navier–Stokes equations

Abstract We consider the Kolmogorov equation associated with the stochastic Navier–Stokes equations in 3D, we prove existence of a solution in the strict or mild sense. The method consists in finding several estimates for the solutions um of the Galerkin approximations of u and their derivatives. These estimates are obtained with the help of an auxiliary Kolmogorov equation with a very irregular negative potential. Although uniqueness is not proved, we are able to construct a transition semigroup for the 3D Navier–Stokes equations. Furthermore, this transition semigroup has a unique invariant measure, which is ergodic and strongly mixing.

Numerical simulation of the stochastic Korteweg-de Vries equation

Abstract In this work, we numerically investigate the influence of a homogeneous noise on the evolution of solitons for the Korteweg–de Vries equation. Our numerical method is based on finite elements and least-squares. We present numerical experiments for different values of noise amplitude and describe different types of behaviours.

Scalar conservation laws with stochastic forcing

We show that the Cauchy Problem for a randomly forced, periodic multi-dimensional scalar first-order conservation law with additive or multiplicative noise is well-posed: it admits a unique solution, characterized by a kinetic formulation of the problem, which is the limit of the solution of the stochastic parabolic approximation.

Hausdorff dimension of a random invariant set

Abstract The notion of random attractor for a dissipative stochastic dynamical system has recently been introduced. It generalizes the concept of global attractor in the deterministic theory. It has been shown that many stochastic dynamical systems associated to a dissipative partial differential equation perturbed by noise do possess a random attractor. In this paper, we prove that, as in the case of the deterministic attractor, the Hausdorff dimension of the random attractor can be estimated by using global Lyapunov exponents. The result is obtained under very natural assumptions. As an application, we consider a stochastic reaction-diffusion equation and show that its random attractor has finite Hausdorff dimension.

Weak approximation of stochastic partial differential equations: the nonlinear case

We study the error of the Euler scheme applied to a stochastic partial dierential equation. We prove that as it is often the case, the weak order of convergence is twice the strong order. A key ingredient in our proof is Malliavin calculus which enables us to get rid of the irregular terms of the error. We apply our method to the case a semilinear stochastic heat equation driven by a space-time white noise.

The Stochastic Nonlinear Schrödinger Equation in H 1

We investigate the local and global existence of solutions in the energy space H 1(R n ) for stochastic nonlinear Schrödinger equations with either additive or multiplicative noise. The noise is assumed to be white in time and correlated in the space variables.

Weak order for the discretization of the stochastic heat equation

In this paper we study the approximation of the distribution of

Hybrid stochastic simplifications for multiscale gene networks

X_t

Convergence of stochastic gene networks to hybrid piecewise deterministic processes.

Hilbert--valued stochastic process solution of a linear parabolic stochastic partial differential equation written in an abstract form as