Stochastic Evolution Equations Driven by a Fractional White Noise

Abstract

In this paper we study stochastic evolution equations driven by a fractional white noise with arbitrary Hurst parameter in infinite dimension. We establish the existence and uniqueness of a mild solution for a nonlinear equation with multiplicative noise under Lipschitz condition by using a fixed point argument in an appropriate inductive limit space. In the linear case with additive noise, a strong solution is obtained. Those results are applied to stochastic parabolic partial differential equations perturbed by a fractional white noise.     

Random attractors for stochastic differential equations driven by two-sided Lévy processes

Abstract

In this paper, the asymptotic behavior of solutions for a nonlinear Marcus stochastic differential equation with multiplicative two-sided Lévy noise is studied. We plan to consider this equation as a random dynamical system. Thus, we have to interpret a Lévy noise as a two-sided metric dynamical system. For that, we have to introduce some fundamental properties of such a noise. So far most studies have only discussed two-sided Lévy processes which are defined by combining two-independent Lévy processes. In this paper, we use another definition of two-sided Lévy process by expanding the probability space. Having this metric dynamical system we will show that the Marcus stochastic differential equation with a particular drift coefficient and multiplicative noise generates a random dynamical system which has a random attractor.     

Attractors for a Damped Stochastic Wave Equation of Sine–Gordon Type with Sublinear Multiplicative Noise

Abstract

The existence of compact random attractors is proved for a damped stochastic wave equation of Sine–Gordon type with sublinear multiplicative noise under homogeneous Dirichlet boundary condition. To be important, in this note a precise estimate of upper bound of Hausdorff dimension of the random attractors is obtained in lower dimension.     

Attractors for stochastic lattice dynamical systems with a multiplicative noise

In this paper, we consider a stochastic lattice differential equation with diffusive nearest neighbor interaction, a dissipative nonlinear reaction term, and multiplicative white noise at each node. We prove the existence of a compact global random attractor which, pulled back, attracts tempered random bounded sets.      

Stochastic reaction diffusion equations on an infinite interval with reflection

We consider a time evolution of random fields with non-negative values on the real line. Such evolution is described by an infinite dimensional stochastic differential equation of Skorokhod's type, which is a stochastic partial differential equation (SPDE) of parabolic type with reflection. We shall show the existence of the solution, and its uniqueness when the diffusion coefficient is constant.     

Local existence and non-explosion of solutions for stochastic fractional partial differential equations driven by multiplicative noise

In this paper we prove the local existence and uniqueness of solutions for a class of stochastic fractional partial differential equations driven by multiplicative noise. We also establish that for this class of equations adding linear multiplicative noise provides a regularizing effect: the solutions will not blow up with high probability if the initial data is sufficiently small, or if the noise coefficient is sufficiently large. As applications our main results are applied to various types of SPDE such as stochastic reaction–diffusion equations, stochastic fractional Burgers equation, stochastic fractional Navier–Stokes equation, stochastic quasi-geostrophic equations and stochastic surface growth PDE.     

Ergodicity for a class of semilinear stochastic partial differential equations

In this paper, we establish the existence and uniqueness of invariant measures for a class of semilinear stochastic partial differential equations driven by multiplicative noise on a bounded domain. The main results can be applied to SPDEs of various types such as the stochastic Burgers equation and the reaction-diffusion equations perturbed by space-time white noise.     

On a random scaled porous media equation

It is shown that a random scaled porous media equation arising from a stochastic porous media equation with linear multiplicative noise through a random transformation is well-posed in L^∞. In the fast diffusion case we show existence in Lp.     

The generalized well-posedness of the Cauchy problem for an abstract stochastic equation with multiplicative noise

We study the existence, uniqueness, and stability of a solution to the Cauchy problem for a stochastic differential equation with multiplicative noise in the spaces of generalized random variables with values in a Hilbert space.     

The Burgers equation with affine linear noise: Dynamics and stability

We study the dynamics of the Burgers equation on the unit interval driven by affine linear noise. Mild solutions of the Burgers stochastic partial differential equation generate a smooth perfect and locally compacting cocycle on the energy space. Using multiplicative ergodic theory techniques, we establish the existence of a discrete non-random Lyapunov spectrum for the cocycle. We establish a local stable manifold theorem near a hyperbolic stationary point, as well as the existence of local smooth invariant manifolds with finite codimension and a countable global invariant foliation of the energy space relative to an ergodic stationary point.     

Stochastic Burgers' equation on the real line: regularity and moment estimates

In this project, we investigate the stochastic Burgers' equation with multiplicative space-time white noise on an unbounded spatial domain. We give a random field solution to this equation by defining a process via a kind of Feynman–Kac representation which solves a stochastic partial differential equation such that its Hopf–Cole transformation solves Burgers' equation. Finally, we obtain Hölder regularity and moment estimates for the solution to Burgers' equation.     

Existence and pathwise uniqueness to an SPDE driven by α-stable colored noise

In this paper we study a stochastic partial differential equation (SPDE) with Hölder continuous coefficient driven by an $\alpha$-stable colored noise. The pathwise uniqueness is proved by using a backward doubly stochastic differential equation backward (SDE) to take care of the Laplacian. The existence of solution is shown by considering the weak limit of a sequence of SDE system which is obtained by replacing the Laplacian operator in the SPDE by its discrete version. We also study an SDE system driven by Poisson random measures.     

Stochastic inertial manifold

A nonlinear stochastic evolution equation in Hilbert space with generalized additive white noise is considered. A concept of stochastic mertial manifold is introduced, defined as a random manifold depending on time, which is finite dimensional, invariant for the dynamic, and attracts exponentially fast all the trajectories as t → ∞. Under the classical spectral gap condition of the deterministic theory, the existence of a stochastic inertial manifold is proved. It is obtained as the solution of a stochastic partial differential equation of degenerate parabolic type, studied by a variant of Bernstein method. A result of existence and uniqueness of a stationary inertial manifold is also proved; the stationary inertial manifold contains the random attractor, introduced in previous works.     

Random attractor of stochastic complex Ginzburg–Landau equation with multiplicative noise on unbounded domain

The existence of a compact random attractor for the stochastic complex Ginzburg–Landau equation with multiplicative noise has been investigated on unbounded domain. The solutions are considered in suitable spaces with weights. According to crucial properties of Ornstein–Uhlenbeck process, using the tail-estimates method, the key uniform a priori estimates for the tail of solutions have been obtained, which give the asymptotic compactness of random attractors. Then the existence of a compact random attractor for the corresponding dynamical system is proved in suitable spaces with weights.     

一类带乘性噪声随机分数阶微分方程Euler方法的弱收敛性与弱稳定性

本文主要在带加性噪声随机分数阶微分方程的基础上,研究了一类更为困难的带乘性噪声随机分数阶微分方程Euler方法的弱收敛性与弱稳定性,并得到了类似的结论.首先构造了数值求解带乘性噪声随机分数阶微分方程的Euler方法,然后证明当分数阶α满足0α1/2时,该方法是1/2-α阶弱收敛的和弱稳定的,文末数值试验的结果验证了理论结果的正确性.     

Stochastic heat equation with random coefficients

We prove the existence of a unique solution for a one-dimensional stochastic parabolic partial differential equation with random and adapted coefficients perturbed by a two-parameter white noise. The proof is based on a maximal inequality for the Skorohod integral deduced from It?'s formula for this anticipating stochastic integral. Received: 21 November 1997 / Revised version: 20 July 1998     

Random attractor and upper semi-continuity for Zakharov lattice system with multiplicative white noise

In this paper, we study the long-term asymptotic behaviour of solutions to the stochastic Zakharov lattice equation with multiplicative white noise. We first transfer the stochastic lattice equation into a random lattice equation and prove the existence and uniqueness of solutions which generate a random dynamical system. Then we consider the existence of a tempered random bounded absorbing set and a random attractor for the system. Finally we establish the upper semi-continuity of random attractor to the global attractor of the limiting system as the coefficients of the white noise terms tend to zero.     

Pathwise Stability of Degenerate Stochastic Evolutions

For linear stochastic evolution equations with linear multiplicative noise, a new method is presented for estimating the pathwise Lyapunov exponent. The method consists of finding a suitable (quadratic) Lyapunov function by means of solving an operator inequality. One of the appealing features of this approach is the possibility to show stabilizing effects of degenerate noise. The results are illustrated by applying them to the examples of a stochastic partial differential equation and a stochastic differential equation with delay. In the case of a stochastic delay differential equation our results improve upon earlier results.     

Existence and Smoothness of the Density for Spatially Homogeneous SPDEs

In this paper, we extend Walsh’s stochastic integral with respect to a Gaussian noise, white in time and with some homogeneous spatial correlation, in order to be able to integrate some random measure-valued processes. This extension turns out to be equivalent to Dalang’s one. Then we study existence and regularity of the density of the probability law for the real-valued mild solution to a general second order stochastic partial differential equation driven by such a noise. For this, we apply the techniques of the Malliavin calculus. Our results apply to the case of the stochastic heat equation in any space dimension and the stochastic wave equation in space dimension d=1,2,3. Moreover, for these particular examples, known results in the literature have been improved.      

Second-order hyperbolic s.p.d.e.'s driven by homogeneous Gaussian noise on a hyperplane

We study a class of hyperbolic stochastic partial differential equations in Euclidean space, that includes the wave equation and the telegraph equation, driven by Gaussian noise concentrated on a hyperplane. The noise is assumed to be white in time but spatially homogeneous within the hyperplane. Two natural notions of solutions are function-valued solutions and random field solutions. For the linear form of the equations, we identify the necessary and sufficient condition on the spectral measure of the spatial covariance for existence of each type of solution, and it turns out that the conditions differ. In spatial dimensions 2 and 3, under the condition for existence of a random field solution to the linear form of the equation, we prove existence and uniqueness of a random field solution to non-linear forms of the equation.