On a stochastic hyperbolic integro-differential equation

In this paper we study an initial-boundary-value problem for a hyperbolic integro-differential equation with random memory and a random noise. We establish the existence, uniqueness and exponential stability of solutions. Our method consists of finite-dimensional approximation and energy estimates.     

Solving a non-linear stochastic pseudo-differential equation of Burgers type

In this paper, we study the initial value problem for a class of non-linear stochastic equations of Burgers type of the following form

_∂tu+q(x,D)u+_∂xf(t,x,u)=_h1(t,x,u)+_h2(t,x,u)_Ft,x${\partial }_{t}u+q(x,D)u+{\partial }_{x}f(t,x,u)=h_1(t,x,u)+h_2(t,x,u)F_{t,x}$

for u:(t,x)∈(0,∞)×R?u(t,x)∈R$u:(t,x)\in (0,\infty )\times \mathbb{R}?u(t,x)\in \mathbb{R}$, where q(x,D)$q(x,D)$ is a pseudo-differential operator with negative definite symbol of variable order which generates a stable-like process with transition density, f,_h1,_h2:[0,∞)×R×R→R$f,h_1,h_2:[0,\infty )\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ are measurable functions, and _Ft,x$F_{t,x}$ stands for a Lévy space-time white noise. We investigate the stochastic equation on the whole space R$\mathbb{R}$ in the mild formulation and show the existence of a unique local mild solution to the initial value problem by utilising a fixed point argument.     

Invariant measure for the stochastic Ginzburg Landau equation

The existence of martingale solutions and stationary solutions of stochastic Ginzburg-Landau equations under general hypothesizes on the dimension, the non linear term and the added noise is investigated. With a few more assumptions, it is established that the transition semi-group is well defined and that the stationary martingale solution yields the existence of an invariant measure. Moreover this invariant measure is shown to be unique.     

On quasi-linear stochastic partial differential equations

Summary We prove existence and uniqueness of the solution of a parabolic SPDE in one space dimension driven by space-time white noise, in the case of a measurable drift and a constant diffusion coefficient, as well as a comparison theorem.and INRIAPartially supported by DRET under contract 901636/A000/DRET/DS/SR     

On the stochastic quasi-linear symmetric hyperbolic system

We establish the existence and uniqueness of a local smooth solution to the Cauchy problem for a quasi-linear symmetric hyperbolic system with random noise in Rd. When the noise is multiplicative satisfying some nondegenerate conditions and the initial data are sufficiently small, we show that the solution exists globally in time in probability, i.e., the probability of global existence can be made arbitrarily close to one if the initial date are small accordingly.     

On non-degenerate quasi-linear stochastic partial differential equations

We prove a limit theorem for non-degenerate quasi-linear parabolic SPDEs driven by space-time white noise in one space-dimension, when the diffusion coefficient is Lipschitz continuous and the nonlinear drift term is only measurable. Hence we obtain an existence and uniqueness and a comparison theorem, which generalize those in [2], [4], [5] to the case of non-degenerate SPDEs with measurable drift and Lipschitz continuous diffusion coefficients.Research supported by the Hungarian National Foundation of Scientific Research No. 2290.     

The 1-d stochastic wave equation driven by a fractional Brownian sheet

In this paper, we develop a Young integration theory in dimension 2 which will allow us to solve a non-linear one- dimensional wave equation driven by an arbitrary signal whose rectangular increments satisfy some Hölder regularity conditions, for some Hölder exponent greater than 1/2. This result will be applied to the fractional Brownian sheet.     

Stochastic conservation laws: Weak-in-time formulation and strong entropy condition

This article is an attempt to complement some recent developments on conservation laws with stochastic forcing. In a pioneering development, Feng and Nualart [8] have developed the entropy solution theory for such problems and the presence of stochastic forcing necessitates introduction of strong entropy condition. However, the authors' formulation of entropy inequalities are weak-in-space but strong-in-time. In the absence of a priori path continuity for the solutions, we take a critical outlook towards this formulation and offer an entropy formulation which is weak-in-time and weak-in-space.     

A phase transition for a stochastic PDE related to the contact process

Summary We consider the one-dimensional heat equation, with a semilinear term and with a nonlinear white noise term. R. Durrett conjectured that this equation arises as a weak limit of the contact process with longrange interactions. We show that our equation possesses a phase transition. To be more precise, we assume that the initial function is nonnegative with bounded total mass. If a certain parameter in the equation is small enough, then the solution dies out to 0 in finite time, with probability 1. If this parameter is large enough, then the solution has a positive probability of never dying out to 0. This result answers a question of Durett.Supported by an NSA grant, and by the Army's Mathematical Sciences Institute at Cornell     

Spatial approximation of stochastic convolutions

We study linear stochastic evolution partial differential equations driven by additive noise. We present a general and flexible framework for representing the infinite dimensional Wiener process, which drives the equation. Since the eigenfunctions and eigenvalues of the covariance operator of the process are usually not available for computations, we propose an expansion in an arbitrary frame. We show how to obtain error estimates when the truncated expansion is used in the equation. For the stochastic heat and wave equations, we combine the truncated expansion with a standard finite element method and derive a priori bounds for the mean square error. Specializing the frame to biorthogonal wavelets in one variable, we show how the hierarchical structure, support and cancelation properties of the primal and dual bases lead to near sparsity and can be used to simplify the simulation of the noise and its update when new terms are added to the expansion.     

Stochastic wave equation of pure jumps: Existence,uniqueness and invariant measures

In this paper, we investigate a class of nonlinear damped stochastic hyperbolic equations with jumps. The jump component considered here is described as a Poisson point process. This paper is divided into two parts. The first part deals with existence and uniqueness of global weak and strong solutions to this type of equations, based on the energy approach. The second part devotes to the existence and support of invariant measures corresponding to the weak solution semi-group, based on Markov property of the solution.     

Periodic-like solutions of variable coefficient and Wick-type stochastic NLS equations

Variable coefficient and Wick-type stochastic nonlinear Schrödinger (NLS) equations are investigated. By using white noise analysis, Hermite transform and extended F-expansion method, we obtain a number of Wick versions of periodic-like wave solutions and periodic wave solutions expressed by various Jacobi elliptic functions for Wick-type stochastic and variable coefficient NLS equations, respectively. In the limit cases, the soliton-like wave solutions are showed as well. Since Wick versions of functions are usually difficult to evaluate, we get some nonWick versions of the solutions for Wick-type stochastic NLS equations in special cases.     

Split-step forward methods for stochastic differential equations

In this paper we discuss split-step forward methods for solving Itô stochastic differential equations (SDEs). Eight fully explicit methods, the drifting split-step Euler (DRSSE) method, the diffused split-step Euler (DISSE) method and the three-stage Milstein (TSM 1a-TSM 1f) methods, are constructed based on Euler-Maruyama method and Milstein method, respectively, in this paper. Their order of strong convergence is proved. The analysis of stability shows that the mean-square stability properties of the methods derived in this paper are improved on the original methods. The numerical results show the effectiveness of these methods in the pathwise approximation of Itô SDEs.     

2D backward stochastic Navier-Stokes equations with nonlinear forcing

The paper is concerned with the existence and uniqueness of a strong solution to a two-dimensional backward stochastic Navier-Stokes equation with nonlinear forcing, driven by a Brownian motion. We use the spectral approximation and the truncation and variational techniques. The methodology features an interactive analysis on the basis of the regularity of the deterministic Navier-Stokes dynamics and the stochastic properties of the Itô-type diffusion processes.     

A Lax equivalence theorem for stochastic differential equations

In this paper, a stochastic mean square version of Lax’s equivalence theorem for Hilbert space valued stochastic differential equations with additive and multiplicative noise is proved. Definitions for consistency, stability, and convergence in mean square of an approximation of a stochastic differential equation are given and it is shown that these notions imply similar results as those known for approximations of deterministic partial differential equations. Examples show that the assumptions made are met by standard approximations.     

Mean-square convergence of stochastic multi-step methods with variable step-size

We study mean-square consistency, stability in the mean-square sense and mean-square convergence of drift-implicit linear multi-step methods with variable step-size for the approximation of the solution of Itô stochastic differential equations. We obtain conditions that depend on the step-size ratios and that ensure mean-square convergence for the special case of adaptive two-step-Maruyama schemes. Further, in the case of small noise we develop a local error analysis with respect to the h$h$–ε$\epsilon$ approach and we construct some stochastic linear multi-step methods with variable step-size that have order 2 behaviour if the noise is small enough.     

A fictitious domain approach to the numerical solution of PDEs in stochastic domains

We present an efficient method for the numerical realization of elliptic PDEs in domains depending on random variables. Domains are bounded, and have finite fluctuations. The key feature is the combination of a fictitious domain approach and a polynomial chaos expansion. The PDE is solved in a larger, fixed domain (the fictitious domain), with the original boundary condition enforced via a Lagrange multiplier acting on a random manifold inside the new domain. A (generalized) Wiener expansion is invoked to convert such a stochastic problem into a deterministic one, depending on an extra set of real variables (the stochastic variables). Discretization is accomplished by standard mixed finite elements in the physical variables and a Galerkin projection method with numerical integration (which coincides with a collocation scheme) in the stochastic variables. A stability and convergence analysis of the method, as well as numerical results, are provided. The convergence is “spectral” in the polynomial chaos order, in any subdomain which does not contain the random boundaries.     

Large deviations and approximations for slow-fast stochastic reaction-diffusion equations

A large deviation principle is derived for a class of stochastic reaction-diffusion partial differential equations with slow-fast components. The result shows that the rate function is exactly that of the averaged equation plus the fluctuating deviation which is a stochastic partial differential equation with small Gaussian perturbation. This result also confirms the effectiveness of the approximation of the averaged equation plus the fluctuating deviation to the slow-fast stochastic partial differential equations.     

Ergodicity for a weakly damped stochastic non-linear Schrödinger equation

We study a damped stochastic non-linear Schr?dinger (NLS) equation driven by an additive noise. It is white in time and smooth in space. Using a coupling method, we establish convergence of the Markov transition semi-group toward a unique invariant probability measure. This kind of method was originally developed to prove exponential mixing for strongly dissipative equations such as the Navier-Stokes equations. We consider here a weakly dissipative equation, the damped nonlinear Schr?dinger equation in the one-dimensional cubic case. We prove that the mixing property holds and that the rate of convergence to equilibrium is at least polynomial of any power.