Comparison Method of Stability Analysis of Semilinear Time-Delay Stochastic Evolution Equation

Abstract

In this paper, we set up the comparison theorem between the mild solution of semilinear time-delay stochastic evolution equation with general time-delay variable and the solution of a class (1-dimension) deterministic functional differential equation, by using the Razumikhin–Lyapunov type functional and the theory of functional differential inequalities. By applying this comparison theorem, we give various types of the stability comparison criteria for the semilinear time-delay stochastic evolution equations. With the aid of these comparison criteria, one can reduce the stability analysis of semilinear time-delay stochastic evolution equations in Hilbert space to that of a class (1-dimension) deterministic functional differential equations. Furthermore, these comparison criteria in special case have been applied to derive sufficient conditions for various stability of the mild solution of semilinear time-delay stochastic evolution equations. Finally, the theories are illustrated with some examples.     

Stability in terms of two measures of solutions to stochastic partial differential delay equations with switching

In this paper, the problem of stability in terms of two measures is considered for a class of stochastic partial differential delay equations with switching. Sufficient conditions for stability in terms of two measures are obtained based on the technique of constructing a proper approximating strong solution system and conducting a limiting type of argument to pass on stability of strong solutions to mild ones. In particular, the stochastic stability under the fixed‐index sequence monotonicity condition and under the average dwell‐time switching are considered.     

ON EXPONENTIAL STABILITY OF NON-AUTONOMOUS STOCHASTIC SEMILINEAR EVOLUTION EQUATIONS

11MroductlonThe purpose ofthls paper Is to Investigate eWone尬lal stability of*theity mild solutions forcenain Hilbert space－Mued stochastlc evoMlon eqll砒ions,Roughy spe出0ng;we cons讪r山efollowing equation:I 伏I＝*x,＋风Il加L十从L,剧dWn,c〔瓜＋咖。(””””“”(11)D 人n 二x．Where A Is the Infinlteslmalgener砒or ofa certain几semigroup S(t),t＞0;on H and F(t;、)and B(t;·)are In general nonlinear mappings from H to H and H to L(x,H),the family ofall bounded linear operators from …     

Stability of semilinear stochastic evolution equations

Stability of moments of the mild solution of a semilinear stochastic evolution equation is studied and sufficient conditions are given for the exponential stability of the pth moment in terms of Liapunov function. Sufficient conditions for sample continuity of the solution are also obtained and the exponential stability of sample paths is proved. Three examples are given to illustrate the theory.     

Mild solutions of evolution quantum stochastic differential equations with nonlocal conditions

We study existence of a unique mild solution of evolution quantum stochastic differential equations with nonlocal conditions under the strong topology. Using the method of successive approximations, we do not need to transform the nonlocal problem to a fixed point form. The evolution operator A generates a family of semigroup that are continuous. Nonlocal conditions allow additional measurements of certain phenomena that cannot be captured by the traditional initial conditions. We show that under some given conditions, the mild solution is unique and also stable. The method applied here is much easier when compared with previous methods used in literature.     

Square-mean almost periodic solutions to some stochastic evolution equations

This paper concerns the square-mean almost periodic mild solutions to a class of abstract nonautonomous functional integro-differential stochastic evolution equations in a real separable Hilbert space. By using the so-called "Acquistapace–Terreni" conditions and the Banach fixed point theorem, we establish the existence, uniqueness and the asymptotical stability of square-mean almost periodic solutions to such nonautonomous stochastic differential equations. As an application, almost periodic solution to a concrete nonautonomous stochastic integro-differential equation is considered to illustrate the applicability of our abstract results.     

Exponential stability for stochastic neutral partial functional differential equations

In this paper, we consider a class of stochastic neutral partial functional differential equations in a real separable Hilbert space. Some conditions on the existence and uniqueness of a mild solution of this class of equations and also the exponential stability of the moments of a mild solution as well as its sample paths are obtained. The known results in Govindan [T.E. Govindan, Almost sure exponential stability for stochastic neutral partial functional differential equations, Stochastics 77 (2005) 139-154], Liu and Truman [K. Liu, A. Truman, A note on almost sure exponential stability for stochastic partial functional differential equations, Statist. Probab. Lett. 50 (2000) 273-278] and Taniguchi [T. Taniguchi, Almost sure exponential stability for stochastic partial functional differential equations, Stoch. Anal. Appl. 16 (1998) 965-975; T. Taniguchi, Asymptotic stability theorems of semilinear stochastic evolution equations in Hilbert spaces, Stochastics 53 (1995) 41-52] are generalized and improved.     

Fixed points and exponential stability of mild solutions of stochastic partial differential equations with delays

The fixed-point theory is first used to consider the stability for stochastic partial differential equations with delays. Some conditions for the exponential stability in pth mean as well as in sample path of mild solutions are given. These conditions do not require the monotone decreasing behavior of the delays, which is necessary in [T. Caraballo, K. Liu, Exponential stability of mild solutions of stochastic partial differential equations with delays, Stoch. Anal. Appl. 17 (1999) 743-763; Ruhollan Jahanipur, Stability of stochastic delay evolution equations with monotone nonlinearity, Stoch. Anal. Appl. 21 (2003) 161-181]. Even in this special case, our results also improve the results in [T. Caraballo, K. Liu, Exponential stability of mild solutions of stochastic partial differential equations with delays, Stoch. Anal. Appl. 17 (1999) 743-763].     

Exponential stability of second-order stochastic evolution equations with Poisson jumps

This paper is concerned with the exponential stability problem of second-order nonlinear stochastic evolution equations with Poisson jumps. By using the stochastic analysis theory, a set of novel sufficient conditions are derived for the exponential stability of mild solutions to the second-order nonlinear stochastic differential equations with infinite delay driven by Poisson jumps. An example is provided to demonstrate the effectiveness of the proposed result.     

Stochastic Retarded Evolution Equations: Green Operators,Convolutions, and Solutions

Abstract

In this work, we shall investigate solution (strong, weak and mild) processes and relevant properties of stochastic convolutions for a class of stochastic retarded differential equations in Hilbert spaces. We introduce a strongly continuous one-parameter family of bounded linear operators which will completely describe the corresponding deterministic systematical dynamics with time delays. This family, which constitutes the fundamental solutions (Green's operators) of our stochastic retarded systems, is applied subsequently to define mild solutions of the stochastic retarded differential equations considered. The relations among strong, weak and mild solutions are explored. By virtue of a strong solution approximation method, Burkholder–Davis–Gundy's type of inequalities for stochastic convolutions are established.     

Exponential stability of mild solutions of stochastic partial differential equations with delays

A semiiinear stochastic partial differential equation with variable delays is considered. Sufficient conditions for the exponential stability in the p-th mean of mild solutions are obtained. Also, pathwise exponential stability is proved. Since the technique ofLyapunov functions is not suitable for delayed equations, the results have been proved by using the properties of the stochastic convolution. As the sufficient conditions obtained are also valid for the case without delays, one can ensure exponential stability of mild solution in some cases where the sufficient conditions in Ichikawa [11] do not give any answer. The results are illustrated with some examples     

非Lipschitz系数下随机泛函微分方程适度解的存在唯一性及其渐近性态(英文)

本文主要运用Picard迭代和算子分数次幂方法,讨论了随机时滞偏微分方程适度解的存在性与唯一性,并对解的渐近性态进行了研究.这里方程的系数不满足Lipschitz条件,时滞r>0为有限的.最后给出了一个非Lipschitz条件的例子.     

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.     

On the Equivalence of Solutions for a Class of Stochastic Evolution Equations in a Banach Space

We study a class of stochastic evolution equations in a Banach space E driven by cylindrical Wiener process. Three different analytical concepts of solutions: generalised strong, weak and mild are defined and the conditions under which they are equivalent are given. We apply this result to prove existence, uniqueness and continuity of weak solutions to stochastic delay evolution equations. We also consider two examples of these equations in non-reflexive Banach spaces: a stochastic transport equation with delay and a stochastic delay McKendrick equation.     

Asymptotic stability of fractional impulsive neutral stochastic partial integro-differential equations with infinite delay

In this article, we investigate the existence and asymptotic stability in p-th moment of a mild solution to a class of neutral stochastic integro-differential equation of fractional order involving non-instantaneous impulses with infinite delay in a Hilbert space. A new set of sufficient conditions proving existence and asymptotic stability of mild solution is derived by utilizing solution operator, functional analysis, stochastic analysis and fixed point technique. Finally, an example is provided to illustrate the obtained abstract result.     

Stability of infinite dimensional stochastic evolution equations with memory and Markovian jumps

A strong solutions approximation approach for mild solutions of stochastic functional differential equations with Markovian switching driven by Lévy martingales in Hilbert spaces is considered. The Razumikhin–Lyapunov type function methods and comparison principles are studied in pursuit of sufficient conditions for the moment exponential stability and almost sure exponential stability of equations in which we are interested. The results of [A.V. Svishchuk, Yu.I. Kazmerchuk, Stability of stochastic delay equations of Itô form with jumps and Markovian switchings, and their applications in finance, Theor. Probab. Math. Statist. 64 (2002) 167–178] are generalized and improved as a special case of our theory.     

Limiting measure and stationarity of solutions to stochastic evolution equations with Volterra noise

Large-time behavior of solutions to stochastic evolution equations driven by two-sided regular cylindrical Volterra processes is studied. The solution is understood in the mild sense and takes values in a separable Hilbert space. Sufficient conditions for the existence of a limiting measure and strict stationarity of the solution process are found and an example for which these conditions are also necessary is provided. The results are further applied to the heat equation perturbed by the two-sided Rosenblatt process.     

Stochastic Differential Equations with Non-Lipschitz Coefficients in Hilbert Spaces

Abstract

In this article, we discuss the successive approximations problem for the solutions of the semilinear stochastic differential equations in Hilbert spaces with cylindrical Wiener processes under some conditions which are weaker than the Lipschitz one. We establish the existence and the uniqueness of the solution and additionally, in our framework we consider a limiting problem for the mild solution. It is shown that the mild solution tends to the solution of the stochastic differential equation of Itô type in finite dimensional space.     

Further results on the existence,uniqueness and stability of strong solutions of quantum stochastic differential equations

Under a more general Lipschitz condition on the coefficients than our consideration in [E.O. Ayoola, Existence and stability results for strong solutions of quantum stochastic differential equations, Stochastic Anal. Appl. 20 (2) (2002) 263–281], we establish the existence, uniqueness and stability of strong solutions of quantum stochastic differential equations (QSDE). This enables us to exhibit a class of Lipschitzian QSDE whose coefficients are continuous on the locally convex space of solution.     

Stability of interconnected stochastic delay systems

A decomposition of a large-scale stochastic functional differential system into a family of isolated subsystems is made. A set of criteria is obtained under which the uniform asymptotic stability with probability one of the trivial solution process of the large-scale system is preserved. These criteria are expressed in terms of the stability of isolated subsystems and bounds on interconnection components. As a tool of our analysis, Razumikhin-type stability conditions for stochastic functional differential systems are also obtained. Several examples are given to show the applicability of our results.