On the discretization of a partial differential equation in the neighborhood of a periodic orbit.
Dynamic programming for the stochastic Navier-Stokes equations
Differentiability of the transition semigroup of the stochastic Burgers equation, and application to the corresponding Hamilton-Jacobi equation
We consider a stochastic Burgers equation. We show that the gradient of the corresponding transition semigroup does exist for any bounded ; and can be estimated by a suitable exponential weight. An application to some Hamilton-Jacobi equation arising in Stochastic Control is given.
On a stochastic Korteweg-de Vries equation with homogeneous noise
Explosion en temps fini pour l’équation de Schrödinger non linéaire stochastique surcritique
Dynamic Programming for the stochastic Navier-Stokes equations
We solve an optimal cost problem for a stochastic Navier-Stokes equation in space dimension 2 by proving existence and uniqueness of a smooth solution of the corresponding Hamilton-Jacobi-Bellman equation.
Optimal control of a stochastic heat equation with boundary-noise and boundary-control
We are concerned with the optimal control of a nonlinear stochastic heat equation on a bounded real interval with Neumann boundary conditions. The specificity here is that both the control and the noise act on the boundary. We start by reformulating the state equation as an infinite dimensional stochastic evolution equation. The first main result of the paper is the proof of existence and uniqueness of a mild solution for the corresponding Hamilton-Jacobi-Bellman (HJB) equation. The ...
Theoretical and numerical aspects of stochastic nonlinear Schrödinger equations
We describe several results obtained recently on stochastic nonlinear Schrödinger equations. We show that under suitable smoothness assumptions on the noise, the nonlinear Schrödinger perturbed by an additive or multiplicative noise is well posed under similar assumptions on the nonlinear term as in the deterministic theory. Then, we restrict our attention to the case of a focusing nonlinearity with critical or supercritical exponent. If the noise is additive, smooth in space and non degenerate,...
Irregular semi-convex gradient systems perturbed by noise and application to the stochastic Cahn–Hilliard equation
Essential m-dissipativity of Kolmogorov operators corresponding to periodic -Navier Stokes equations
We prove the essential m-dissipativity of the Kolmogorov operator associated with the stochastic Navier-Stokes flow with periodic boundary conditions in a space where is an invariant measure
Soliton dynamics for the Korteweg-de Vries equation with multiplicative homogeneous noise.
A modified Kardar-Parisi-Zhang model.
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