121 research outputs found
On the stochastic Strichartz estimates and the stochastic nonlinear Schr\"odinger equation on a compact riemannian manifold
We prove the existence and the uniqueness of a solution to the stochastic
NSLE on a two-dimensional compact riemannian manifold. Thus we generalize a
recent work by Burq, G\'erard and Tzvetkov in the deterministic setting, and a
series of papers by de Bouard and Debussche, who have examined similar
questions in the case of the flat euclidean space with random perturbation. We
prove the existence and the uniqueness of a local maximal solution to
stochastic nonlinear Schr\"odinger equations with multiplicative noise on a
compact d-dimensional riemannian manifold. Under more regularity on the noise,
we prove that the solution is global when the nonlinearity is of defocusing or
of focusing type, d=2 and the initial data belongs to the finite energy space.
Our proof is based on improved stochastic Strichartz inequalities
Rate of Convergence of Implicit Approximations for stochastic evolution equations
Stochastic evolution equations in Banach spaces with unbounded nonlinear
drift and diffusion operators are considered. Under some regularity condition
assumed for the solution, the rate of convergence of implicit Euler
approximations is estimated under strong monotonicity and Lipschitz conditions.
The results are applied to a class of quasilinear stochastic PDEs of parabolic
type.Comment: 25 page
Large deviation principle and inviscid shell models
A LDP is proved for the inviscid shell model of turbulence. As the viscosity
coefficient converges to 0 and the noise intensity is multiplied by the square
root of the viscosity, we prove that some shell models of turbulence with a
multiplicative stochastic perturbation driven by a H-valued Brownian motion
satisfy a LDP in C([0,T],V) for the topology of uniform convergence on [0,T],
but where V is endowed with a topology weaker than the natural one. The initial
condition has to belong to V and the proof is based on the weak convergence of
a family of stochastic control equations. The rate function is described in
terms of the solution to the inviscid equation
Large deviations for rough paths of the fractional Brownian motion
Starting from the construction of a geometric rough path associated with a
fractional Brownian motion with Hurst parameter given by
Coutin and Qian (2002), 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.Comment: 32 page
On discretization schemes for stochastic evolution equations
International audienceStochastic 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
On the stochastic Strichartz estimates and the stochastic nonlinear Schrödinger equation on a compact riemannian manifold
International audienceWe prove the existence and the uniqueness of a solution to the stochastic NSLE on a two-dimensional compact riemannian manifold. Thus we generalize a recent work by Burq, Gérard and Tzvetkov in the deterministic setting, and a series of papers by de Bouard and Debussche, who have examined similar questions in the case of the flat euclidean space with random perturbation. We prove the existence and the uniqueness of a local maximal solution to stochastic nonlinear Schrödinger equations with multiplicative noise on a compact d-dimensional riemannian manifold. Under more regularity on the noise, we prove that the solution is global when the nonlinearity is of defocusing or of focusing type, d=2 and the initial data belongs to the finite energy space. Our proof is based on improved stochastic Strichartz inequalities
Stochastic 2D hydrodynamical systems: Wong-Zakai approximation and Support theorem
We deal with a class of abstract nonlinear stochastic models with
multiplicative noise, which covers many 2D hydrodynamical models including the
2D Navier-Stokes equations, 2D MHD models and 2D magnetic B\'enard problems as
well as some shell models of turbulence. Our main result describes the support
of the distribution of solutions. Both inclusions are proved by means of a
general Wong-Zakai type result of convergence in probability for non linear
stochastic PDEs driven by a Hilbert-valued Brownian motion and some adapted
finite dimensional approximation of this process
Strong L2 convergence of time Euler schemes for stochastic 3D Brinkman-Forchheimer-Navier-Stokes equations
We prove that some time Euler schemes for the 3D Navier-Stokes equations
modified by adding a Brinkman-Forchheimer term and a random perturbation
converge in . This extends previous results concerning the strong
rate of convergence of some time discretization schemes for the 2D Navier
Stokes equations. Unlike the 2D case, our proposed 3D model with the
Brinkman-Forchheimer term allows for a strong rate of convergence of order
almost 1/2, that is independent of the viscosity parameter
Accelerated finite elements schemes for parabolic stochastic partial differential equations
For a class of finite elements approximations for linear stochastic parabolic
PDEs it is proved that one can accelerate the rate of convergence by Richardson
extrapolation. More precisely, by taking appropriate mixtures of finite
elements approximations one can accelerate the convergence to any given speed
provided the coefficients, the initial and free data are sufficiently smooth.Comment: 1 figur
On the splitting method for some complex-valued quasilinear equations
Using the approach of the splitting method developed by I. Gy\"ongy and N.
Krylov for parabolic quasi linear equations, we study the speed of convergence
for general complex-valued stochastic evolution equations. The approximation is
given in general Sobolev spaces and the model considered here contains both the
parabolic quasi-linear equations under some (non strict) stochastic
parabolicity condition as well as linear Schr\"odinger equation
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