Arnaud Debussche

Arnaud Debussche

We study the error of the Euler scheme applied to a stochastic partial differential 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.

Arnaud Debussche, Jacques Printems

In this paper we study the approximation of the distribution of $X_t$ Hilbert--valued stochastic process solution of a linear parabolic stochastic partial differential equation written in an abstract form as $$ dX_t+AX_t dt = Q^{1/2} d W_t, \quad X_0=x \in H, \quad t\in[0,T], $$ driven by a Gaussian space time noise whose covariance operator $Q$ is given. We assume that $A^{-α}$ is a finite trace operator for some $α>0$ and that $Q$ is bounded from $H$ into $D(A^β)$ for some $β\geq 0$. It is not required to be nuclear or to commute with $A$. The discretization is achieved thanks to finite element methods in space (parameter $h>0$) and implicit Euler schemes in time (parameter $Δt=T/N$). We define a discrete solution $X^n_h$ and for suitable functions $ϕ$ defined on $H$, we show that $$ |\E ϕ(X^N_h) - \E ϕ(X_T) | = O(h^{2γ} + Δt^γ) $$ \noindent where $γ<1- α+ β$. Let us note that as in the finite dimensional case the rate of convergence is twice the one for pathwise approximations.

Arnaud Debussche, Erwan Faou

We consider numerical approximations of stochastic differential equations by the Euler method. In the case where the SDE is elliptic or hypoelliptic, we show a weak backward error analysis result in the sense that the generator associated with the numerical solution coincides with the solution of a modified Kolmogorov equation up to high order terms with respect to the stepsize. This implies that every invariant measure of the numerical scheme is close to a modified invariant measure obtained by asymptotic expansion. Moreover, we prove that, up to negligible terms, the dynamic associated with the Euler scheme is exponentially mixing.

Arnaud Debussche, Sylvain De Moor, Martina Hofmanova

We consider a quasilinear parabolic stochastic partial differential equation driven by a multiplicative noise and study regularity properties of its weak solution satisfying classical a priori estimates. In particular, we determine conditions on coefficients and initial data under which the weak solution is Hölder continuous in time and possesses spatial regularity that is only limited by the regularity of the given data. Our proof is based on an efficient method of increasing regularity: the solution is rewritten as the sum of two processes, one solves a linear parabolic SPDE with the same noise term as the original model problem whereas the other solves a linear parabolic PDE with random coefficients. This way, the required regularity can be achieved by repeatedly making use of known techniques for stochastic convolutions and deterministic PDEs.

Arnaud Debussche, Erwan Faou

We consider the linear Schrödinger equation and its discretization by split-step methods where the part corresponding to the Laplace operator is approximated by the midpoint rule. We show that the numerical solution coincides with the exact solution of a modified partial differential equation at each time step. This shows the existence of a modified energy preserved by the numerical scheme. This energy is close to the exact energy if the numerical solution is smooth. As a consequence, we give uniform regularity estimates for the numerical solution over arbitrary long time

Arnaud Debussche, Cyril Odasso

We construct a Markov family of solutions for the 3D Navier-Stokes equation perturbed by a non degenerate noise. We improve the result of [DPD-NS3D] in two directions. We see that in fact not only a transition semigroup but a Markov family of solutions can be constructed. Moreover, we consider a state dependant noise. Another feature of this work is that we greatly simplify the proofs of [DPD-NS3D].