The Limit Distribution of Stochastic Evolution Equations Driven by-Stable Non-Gaussian Noise

We study the distribution limit of a class of stochastic evolution equation driven by an additive-stable Non-Gaussian process in the case of α∈(1,2).We prove that,under suitable conditions,the law of the solution converges weakly to the law of a stochastic evolution equation with an additive Gaussian process.

CONTROLLABILITY OF NEUTRAL STOCHASTIC EVOLUTION EQUATIONS DRIVEN BY FRACTIONAL BROWNIAN MOTION

In this paper,we investigate the controllability for neutral stochastic evolution equations driven by fractional Brownian motion with Hurst parameter H ∈（1/2,1） in a Hilbert space.We employ the α-norm in order to reflect the relationship between H and the fractional power α.Sufficient conditions are established by using stochastic analysis theory and operator theory.An example is provided to illustrate the effectiveness of the proposed result.

Extended Riccati Equation Rational Expansion Method and Its Application to Nonlinear Stochastic Evolution Equations

In this work, by means of a new more general ansatz and the symbolic computation system Maple, we extend the Riccati equation rational expansion method [Chaos, Solitons ＆ Fractals 25 （2005） 1019] to uniformly construct a series of stochastic nontravelling wave solutions for nonlinear stochastic evolution equation. To illustrate the effectiveness of our method, we take the stochastic mKdV equation as an example, and successfully construct some new and more general solutions including a series of rational formal nontraveling wave and coefficient functions＇ soliton-like solution.s and trigonometric-like function solutions. The method can also be applied to solve other nonlinear stochastic evolution equation or equations.

Freidlin-Wentzell’s Large Deviations for Stochastic Evolution Equations with Poisson Jumps

We establish a Freidlin-Wentzell’s large deviation principle for general stochastic evolution equations with Poisson jumps and small multiplicative noises by using weak convergence method.

Square-mean Almost Automorphic Solutions to Some Stochastic Evolution Equations I: Autonomous Case

This paper concerns the square-mean almost automorphic solutions to a class of abstract semilinear functional integro-differential stochastic evolution equations in real separable Hilbert spaces. Under some suitable assumptions, the existence, uniqueness and asymptotic stability of the square-mean almost automorphic mild solution to some stochastic differential equations are established. As an application, we analyze the almost automorphic mild solution to some stochastic partial functional differential equation which turns out to be in good agreement with our abstract results.

Approximating solutions of neutral stochastic evolution equations with jumps

In this paper, we establish existence and uniqueness of the mild solutions to a class of neutral stochastic evolution equations driven by Poisson random measures in some Hilbert space. Moreover, we adopt the Faedo-Galerkin scheme to approximate the solutions.

Exit problems for nonlinear stochastic evolution equations on Hilbert spaces

This paper extends exit theorems of Da Prato and Zabczyk to nonconstant diffusion coefficients.It uses extensively general, exponential estimates due to Peszat.

ON EXPONENTIAL STABILITY OF NON-AUTONOMOUS STOCHASTIC SEMILINEAR EVOLUTION EQUATIONS

Sufficient conditions for the exponential stability of a class of nonlinear, non-autonomous stochastic differential equations in infinite dimensions are studied. The analysis consists of introducing a suitable approximating solution systems and usig a limiting argument to pass on stability of strong solutions to mild ones. Consequently, under these conditions the random attractors of given stochastic systems are reduced to zero with exponential decay. Lastly, two examples are investigated to illustrate the theory.

THE PATHWISE SOLUTION FOR A CLASS OF QUASILINEAR STOCHASTIC EQUATIONS OF EVOLUTION IN BANACH SPACE Ⅲ

This is the third part of the papers with the same title. We will discuss the problem of convergence of the semi-implicit difference scheme for a class of quasilinear SEE, which generalize the Crandall's work to the stochastic case.

A variational formula for controlled backward stochastic partial differential equations and some application

An optimal control problem for a controlled backward stochastic partial differential equation in the abstract evolution form with a Bolza type performance functional is considered. The control domain is not assumed to be convex, and all coefficients of the system are allowed to be random. A variational formula for the functional in a given control process direction is derived, by the Hamiltonian and associated adjoint system. As an application, a global stochastic maximum principle of Pontraygins type for the optimal controls is established.

UNIQUENESS OF THE MILD SOLUTION OF SEMILINEAR STOCHASTIC EVOLUTION EQUATION IN HILBERT SPACE

In this paper, we will consider following initial value problem of semilinear stochastic evolution equation in Hilbert Space: [GRAPHICS] where W(t) is a wiener process in H, H and Y are two real separable Hilbert Spaces, A is an infinitesimal generator of a strongly continuous semigroup s(t) on Y, f(t, y): [0, T] x Y --> Y, and G(t, y): [0, T] X Y --> L(H, Y), y0: OMEGA --> Y is a ramdom variable of square integrable. We apply theory of the semigroup and obtain two conclusions of uniqueness of the mild solution of (1) which include the corresponding results in [4].

Extended Jacobi Elliptic Function Rational Expansion Method and Its Application to （2＋1）-Dimensional Stochastic Dispersive Long Wave System

In this work, by means of a generalized method and symbolic computation, we extend the Jacobi elliptic function rational expansion method to uniformly construct a series of stochastic wave solutions for stochastic evolution equations. To illustrate the effectiveness of our method, we take the （2＋ 1）-dimensional stochastic dispersive long wave system as an example. We not only have obtained some known solutions, but also have constructed some new rational formal stochastic Jacobi elliptic function solutions.

Error estimates of a finite element method for stochastic time-fractional evolution equations with fractional Brownian motion

The aim of this paper is to consider the convergence of the numerical methods for stochastic time-fractional evolution equations driven by fractional Brownian motion.The spatial and temporal regularity of the mild solution is given.The numerical scheme approximates the problem in space by the Galerkin finite element method and in time by the backward Euler convolution quadrature formula,and the noise by the L 2-projection.The strong convergence error estimates for both semi-discrete and fully discrete schemes are established.A numerical example is presented to verify our theoretical analysis.

Mild Solution of Stochastic Equations with Levy Jumps： Existence, Uniqueness, Regularity and Stability

The existence and uniqueness of mild solution to stochastic equations with jumps are established, a stochastic Fubini theorem and a type of Burkholder-Davis- Gundy inequality are proved, and the two formulas are used to study the regularity property of the mild solution of a general stochastic evolution equation perturbed by Levy process. Then the authors prove the moment exponential stability, almost sure exponential stability and comparison principles of the mild solution. As applications, the stability and comparison principles of stochastic heat equation with Levy jump are given.

On Small Time Large Deviation Principle for Diffusion Processes on Hilbert Spaces under Non-Lipschitzian Condition

Under the non-Lipschitzian condition, a small time large deviation principle of diffusion processes on Hilbert spaces is established. The operator theory and Gronwall inequality play an important role.