Annie Millet

Stochastic 2D Hydrodynamical Type Systems: Well Posedness and Large Deviations

We deal with a class of abstract nonlinear stochastic models, which covers many 2D hydrodynamical models including 2D Navier-Stokes equations, 2D MHD models and the 2D magnetic Bénard problem and also some shell models of turbulence. We state the existence and uniqueness theorem for the class considered. Our main result is a Wentzell-Freidlin type large deviation principle for small multiplicative noise which we prove by a weak convergence method.

On Discretization Schemes for Stochastic Evolution Equations

Abstract Stochastic evolutional equations with monotone operators are considered in Banach spaces. Explicit and implicit numerical schemes are presented. The convergence of the approximations to the solution of the equations is proved.

Uniform large deviations for parabolic SPDEs and applications

Let denote the set of functions f(t,x) which are [alpha]-Holder continuous in t and 2[alpha]-Holder continuous in x. For 0

Large deviations for the Boussinesq equations under random influences

A Boussinesq model for the Benard convection under random influences is considered as a system of stochastic partial differential equations. This is a coupled system of stochastic Navier-Stokes equations and the transport equation for temperature. Large deviations are proved, using a weak convergence approach based on a variational representation of functionals of infinite-dimensional Brownian motion.

Rate of Convergence of Space Time Approximations for Stochastic Evolution Equations

Stochastic evolution equations in Banach spaces with unbounded nonlinear drift and diffusion operators driven by a finite dimensional Brownian motion are considered. Under some regularity condition assumed for the solution, the rates of convergence of various numerical approximations are estimated under strong monotonicity and Lipschitz conditions. The abstract setting involves general consistency conditions and is then applied to a class of quasilinear stochastic PDEs of parabolic type.

Large deviations for rough paths of the fractional Brownian motion

Abstract Starting from the construction of a geometric rough path associated with a fractional Brownian motion with Hurst parameter H ∈ ] 1 4 , 1 2 [ given by Coutin and Qian in [Probab. Theory Related Fields 122 (2002) 108–140], we prove a large deviation principle in the space of geometric rough paths, extending classical results on Gaussian processes. As a by-product, geometric rough paths associated to elements of the reproducing kernel Hilbert space of the fractional Brownian motion are obtained and an explicit integral representation is given.

Approximation and support theorem for a wave equation in two space dimensions

We prove a characterization of the support of the law of the solution for a stochastic wave equation with two-dimensional space variable, driven by a noise white in time and correlated in space. The result is a consequence of an approximation theorem, in the convergence of probability, for equations obtained by smoothing the random noise. For some particular classes of coefficients, approximation in the L p -norm for p > 1 is also proved.

On implicit and explicit discretization schemes for parabolic SPDEs in any dimension

We study the speed of convergence of the explicit and implicit space-time discretization schemes of the solution u(t,x) to a parabolic partial differential equation in any dimension perturbed by a space-correlated Gaussian noise. The coefficients only depend on u(t,x) and the influence of the correlation on the speed is observed.

On a stochastic wave equation in two space dimensions: regularity of the solution and its density

We pursue the investigation started in a recent paper by Millet and Sanz-Sole (1999, Ann. Probab. 27, 803-844) concerning a non-linear wave equation driven by a Gaussian white noise in time and correlated in the two-dimensional space variable. Under more restrictive conditions on the covariance function of the noise, we prove Holder-regularity properties for both the solution and its density. For the latter, we adapt the method used in a paper by Morien (1999, Bernoulli: Official J. Bernoulli Soc. 5(2), 275-298) based on the Malliavin calculus.

The support of the solution to a hyperbolic SPDE

SummaryIn this paper we prove Stroock-Varadhan type theorems for the topological support of a hyperbolic stochastic partial differential equation in the α-Hölder norm, for α∈(0, 1/2). Our approach is based on absolutely continuous transformations of Ω defined using non-homogeneous approximations of the Brownian sheet.