Stochastic Partial Functional Differential Equations Research Articles

Pseudo almost automorphy of stochastic neutral partial functional differential equations with Lévy noise

ABSTRACT The paper introduces and studies the concepts of (Poisson) -pseudo almost automorphy of class r for stochastic processes. Under suitable conditions on the coefficients, we establish the existence, and uniqueness of mild solutions, which are -pseudo almost automorphic of class r in distribution for stochastic neutral partial functional differential equations with Lévy noise.

Trotter-Kato approximations of stochastic neutral partial functional differential equations

Trotter-Kato approximations of stochastic neutral partial functional differential equations

Averaging of neutral stochastic partial functional differential equations involving delayed impulses

In this paper, we initiate a study on averaging principles for neutral stochastic partial functional differential equations (NSPFDEs) with delayed impulses. With the aid of inequality techniques, the semigroup approach and some technical transformations, we derive an averaging principle for a class of NSPFDEs involving delayed impulses. The results obtained ensure that one can focus on the averaged autonomous system without impulses in place of the original system, thus providing an effective way for reducing complexity. In the end, an example is worked out to demonstrate the theoretical results.

Global attracting set, exponential stability and stability in distribution of SPDEs with jumps

A novel approach to the global attracting sets of mild solutions for stochastic functional partial differential equations driven by Lévy noise is presented. Consequently, some new sufficient conditions ensuring the existence of the global attracting sets of mild solutions for the considered equations are established. As applications, some new criteria for the exponential stability in mean square of the considered equations is obtained. Subsequently, by employing a weak convergence approach, we try to establish some stability conditions in distribution of the segment processes of mild solutions to stochastic delay partial differential equations with jumps under some weak conditions. Some known results are improved. Lastly, some examples are investigated to illustrate the theory.

Global attracting sets and exponential stability of stochastic partial functional differential equations

This paper is concerned with the global attracting sets and the exponential stability in the mean square of mild solutions for stochastic functional partial differential equations. Some new sufficient conditions ensuring the existence of the global attracting sets and the exponential stability in the mean square of mild solutions for the considered equations are established by introducing approximating systems with strong solutions and using a limiting argument. Finally, some examples are investigated to illustrate the theory.

Existence, Uniqueness and Stability of Impulsive Stochastic Partial Neutral Functional Differential Equations with Infinite Delays Driven by a Fractional Brownian Motion

This article presents the result on existence, uniqueness and stability of mild solution of impulsive stochastic partial neutral functional differential equations driven by a fractional Brownian motion. The results are obtained by using the method of successive approximation and Bihari’s inequality.

Existence and Stability Results of Impulsive Stochastic Partial Neutral Functional Differential Equations with Infinite Delays and Poisson Jumps

In this paper, we are focused upon the results on existence, uniqueness and stability of mild solution of impulsive stochastic partial neutral functional differential equations with infinite delays and poisson jumps. The results are obtained by using the method of successive approximation and Bihari’s inequality.

Harnack and Shift Harnack Inequalities for Degenerate (Functional) Stochastic Partial Differential Equations with Singular Drifts

The existence and uniqueness of the mild solutions for a class of degenerate functional stochastic partial differential equations (SPDEs) are obtained, where the drift is assumed to be Holder–Dini continuous. Moreover, the non-explosion of the solution is proved under some reasonable conditions. In addition, the Harnack inequality is derived by the method of coupling by change of measure. Finally, the shift Harnack inequality is obtained for the equations without delay, which is new even in the non-degenerate case. An example is presented in the final part of the paper.

Stability of a Class of Impulsive Neutral Stochastic Functional Partial Differential Equations

In this paper, a class of impulsive neutral stochastic functional partial differential equations driven by Brownian motion and fractional Brownian motion is investigated. Under some suitable assumptions, the pth moment exponential stability is discussed by means of the fixed-point theorem. Our results also improve and generalize some previous studies. Moreover, one example is given to illustrate our main results.

Exponential stability for neutral stochastic functional partial differential equations driven by Brownian motion and fractional Brownian motion

In this paper, we study the exponential stability in the pth moment of mild solutions to neutral stochastic functional partial differential equations driven by Brownian motion and fractional Brownian motion: \t\t\td[x(t)+g(t,xt)]=[Ax(t)+f(t,xt)]dt+h(t,xt)dW(t)+σ(t)dBH(t),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ d \\bigl[x(t)+g(t,x_{t}) \\bigr]= \\bigl[Ax(t)+f(t,x_{t}) \\bigr] \\,dt+h(t,x_{t})\\,dW(t)+\\sigma(t)\\,dB^{H}(t), $$\\end{document} where Hin(1/2,1). Our method for investigating the stability of solutions is based on the Banach fixed point theorem. The obtained results generalize and improve the results due to Boufoussi and Hajji (Stat. Probab. Lett. 82:1549–1558, 2012), Caraballo et al. (Nonlinear Anal. 74:3671–3684, 2011), and Luo (J. Math. Anal. Appl. 355:414–425, 2009).

Well-Posedness of Mild Solutions to Stochastic Parabolic Partial Functional Differential Equations

In this paper, we study the well-posedness of mild solutions to stochastic parabolic partial functional differential equations with space–time white noise. Firstly, we establish an existence–uniqueness theorem under the global Lipschitz condition and the linear growth condition. Secondly, we show the existence–uniqueness property under the global/local Lipschitz condition but without assuming the linear growth condition. In particular, we consider the existence and uniqueness under the weaker condition than the Lipschitz condition. Finally, we obtain the nonnegativity and comparison theorems and utilize them to investigate the existence of nonnegative mild solutions under the linear growth condition without assuming the Lipschitz condition.

Mild Solutions and Harnack Inequality for Functional Stochastic Partial Differential Equations with Dini Drift

The existence and uniqueness of a mild solution for a class of functional stochastic partial differential equations with multiplicative noise and a locally Dini continuous drift are proved. In addition, under a reasonable condition the solution is non-explosive. Moreover, Harnack inequalities are derived for the associated semigroup under certain global conditions, which is new even in the case without delay.

English

We apply an approximation by means of the method of lines for hyperbolic stochastic functional partial differential equations driven by one-dimensional Brownian motion. We study the stability with respect to small L2-perturbations.

Stationary solutions of neutral stochastic partial differential equations with delays in the highest-order derivatives

In this work, we shall consider the existence and uniqueness of stationary solutions to stochastic partial functional differential equations with additive noise in which a neutral type of delay is explicitly presented. We are especially concerned about those delays appearing in both spatial and temporal derivative terms in which the coefficient operator under spatial variables may take the same form as the infinitesimal generator of the equation. We establish the stationary property of the neutral system under investigation by focusing on distributed delays. In the end, an illustrative example is analyzed to explain the theory in this work.

Asymptotic behavior, attracting and quasi-invariant sets for impulsive neutral SPFDE driven by Lévy noise

By establishing two new impulsive-integral inequalities, the attracting and quasi-invariant sets of the mild solution for impulsive neutral stochastic partial functional differential equations driven by Lévy noise are obtained, respectively. Moreover, we shall derive some sufficient conditions to ensure stability of this mild solution in the sense of both moment exponential stability and almost surely exponential stability.

The asymptotic behavior for neutral stochastic partial functional differential equations

ABSTRACTIn this article, we consider the existence and uniqueness, and the asymptotic behavior of the strong solution for the following neutral stochastic partial functional differential equations.where A(t): V → V* is a linear bounded operator. , and are some appropriate measurable functions. By establishing the variational framework, the existence and uniqueness for the strong solution of such equations is shown under the coercivity condition. Then, the asymptotic behavior of the strong solution is investigated and some well-known results are extended. As a by-product, the exponential stability in mean square and almost sure exponential stability for a strong solution of neutral stochastic partial differential equations with delays are also discussed by utilizing the integral inequality. Finally, an illustrative example is given to demonstrate the effectiveness of the obtained results.

Averaging principles for functional stochastic partial differential equations driven by a fractional Brownian motion modulated by two-time-scale Markovian switching processes

Motivated by applications of hybrid systems, this work considers functional stochastic partial differential equations (FSPDEs) driven by a fractional Brownian motion (fBm) modulated by a two-time-scale Markov chain with a finite state space. Our aim is to obtain an averaging principle for such systems with fast–slow Markov switching processes. Under suitable conditions, it is proved that there is a limit process in which the fast changing “noise” is averaged out and the limit is an average with respect to the stationary measure of the fast-varying processes. The limit process, being substantially simpler than that of the original system, can be used to reduce the computational complexity. There are several difficulties in our problems. First, because of the use of fBm, the techniques of martingale problem formulation can no longer be used. Second, there is no strong solution available and the underlying FSPDEs admit only a unique mild solution. Moreover, although the regime-switching enlarges the applicability of the underlying systems, to treat such systems is more difficult. To overcome the difficulties, fixed point theorem together with the use of stopping time argument, and a semigroup approach are used.

Existence and exponential stability for impulsive stochastic partial functional differential equations

In this paper, the existence and uniqueness, the exponential stability, and the almost sure exponential stability of mild solution for impulsive stochastic partial functional differential equations with finite delay are considered. Some sufficient conditions are established for our concerned problems, and some existing results are generalized and improved. Finally, an illustrative example is provided to show the feasibility and effectiveness of the obtained results.

English

english

Stability of second-order stochastic neutral partial functional differential equations driven by impulsive noises

This paper investigates the stability of second-order stochastic neutral partial functional differential equations driven by impulsive noises. Some sufficient conditions ensuring $p$th moment exponential stability of the second-order stochastic neutral partial functional differential equations driven by impulsive noises are obtained by establishing a new impulsive-integral inequality. These existing results are generalized and improved by the present study. Finally, an example is given to show the effectiveness of our results.