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This paper studies a class of McKean-Vlasov stochastic functional differential equations in which the dynamics of the underlying process may depend on its history as well as the distribution of the history. These stochastic processes provide a general and versatile framework for modeling and analyzing complex systems with inherent distribution- and memory-dependent dynamical behaviors. Under mild conditions, this paper establishes the well-posedness for the associated McKean-Vlasov stochastic functional differential equations and derives a propagation of chaos as well as a continuity with respect to the initial distribution result. Several examples are studied as an illustration. [ABSTRACT FROM AUTHOR]

This work focuses on stochastic functional differential equations (SFDEs) with wide-band noise perturbation, which is a class of actual physical process and has wide application in the modeling of communication and signal system. Using functional derivatives together with martingale methods and weak convergence techniques, this paper examines the asymptotic properties of the underlying systems as the small parameter tends to zero. Based on the techniques, this paper also establishes the average principle for a class of SFDEs with two-time scales. As a special case, this paper also considers a class of integro-differential equations with wide-band noise perturbation and examines its approximate properties. As an example, a scalar stochastic Lotka–Volterra integro-differential system with wide-band noise perturbation is investigated. [ABSTRACT FROM AUTHOR]

This paper examines the stationary distribution of the stochastic Lotka-Volterra model with infinite delay. Since the solutions of stochastic functional or delay differential equations depend on their history, they are non-Markov, which implies that traditional techniques based on the Markov property, cannot be applicable. This paper uses the variable substituting technique to obtain a higher-dimensional stochastic differential equation without delay. In this paper it is shown that the new equation satisfies the strong Feller property and strong Markov property, by which the invariant measure follows from the Krylov-Bogoliubov Theorem. Since distribution of the original model is actually the marginal distribution of the new equation, these results also show that there exists the stationary distribution for the original equation. Moreover, when the noise intensity is strongly dependent on the population size, by Hörmander's theorem, this paper also shows that the stationary distribution of this stochastic Lotka-Volterra system with infinite delay holds a C ∞ -smooth density. [ABSTRACT FROM AUTHOR]

Using Malliavin calculus, this paper establishes asymptotic Bismut formulae for stochastic functional differential equations with infinite delay. Both nondegenerate and degenerate diffusion coefficients are treated. In addition, combined with the corresponding exponential ergodicity, stabilization bounds for \nabla P_{t}f as t\rightarrow \infty are derived. [ABSTRACT FROM AUTHOR]

This paper focuses on a class of stochastic functional differential equations with infinite delay and non-Lipschitz coefficients. Under one-sided super-linear growth and non-Lipschitz conditions, this paper establishes the existence and uniqueness of strong solutions and strong Markov properties of the segment processes. Under additional assumption on non-degeneracy of the diffusion coefficient, exponential ergodicity for the segment process is derived by using asymptotic coupling method. In addition, the asymptotic log-Harnack inequality is established for the associated Markovian semigroup by using coupling and change of measures, which implies the asymptotically strong Feller property. Finally, an example is given to demonstrate these results. [ABSTRACT FROM AUTHOR]

This paper focuses on two-time-scale coupled stochastic functional differential equations (SFDEs). The system under consideration has a slow component and a fast component. Both components depend on the segment process (an infinite dimension process) of the slow component. To overcome the difficulty due to the past dependence and the coupling of the segment process, such properties as the Hölder continuity and tightness on a space of continuous functions are investigated first for the segment process. In addition, it is also shown that the solution of a fixed- x equation depends continuously on the parameters. Then using the martingale problem formulation, an average principle is established by a direct averaging. [ABSTRACT FROM AUTHOR]

This paper examines limit theorems for a class of stochastic functional differential equations with infinite delay. Under non-Lipschitz conditions, the strong law of large numbers and the central limit theorem for additive functionals of the segment processes are established by using limit theorems of uniform mixing Markovian processes. Under some dissipative assumptions, the law of iterated logarithm is also derived for additive functionals of such equations via a martingale difference sequence of square integrable martingales. [ABSTRACT FROM AUTHOR]

This paper focuses on a class of regime-switching functional diffusion processes with infinite delay and develops a central limit theorem (CLT) for additive functionals under uniform mixing conditions. In addition, a law of iterated logarithm (LIL) for the additive functionals is also established by using the square integrable martingale difference sequences. Finally, two examples are given to illustrate our results. [ABSTRACT FROM AUTHOR]

Delays are ubiquitous, pervasive, and entrenched in everyday life, thus taking it into consideration is necessary. Dupire recently developed a functional Itô formula, which has changed the landscape of the study of stochastic functional differential equations and encouraged a reconsideration of many problems and applications. Based on the new development, this work examines functional diffusions with two-time scales in which the slow-varying process includes path-dependent functionals and the fast-varying process is a rapidly-changing diffusion. The gene expression of biochemical reactions occurring in living cells in the introduction of this paper is such a motivating example. This paper establishes mixed functional Itô formulas and the corresponding martingale representation. Then it develops an averaging principle using weak convergence methods. By treating the fast-varying process as a random "noise", under appropriate conditions, it is shown that the slow-varying process converges weakly to a stochastic functional differential equation whose coefficients are averages of that of the original slow-varying process with respect to the invariant measure of the fast-varying process. [ABSTRACT FROM AUTHOR]

This paper considers stochastic functional differential equations (SFDEs) with infinite delay. The main aim is to establish the LaSalle-type theorems to locate limit sets for this class of SFDEs. In comparison with the existing results, this paper gives more general results under the weaker conditions imposed on the Lyapunov function. These results can be used to discuss the asymptotic stability and asymptotic boundedness for SFDEs with infinite delay. In the end, two examples will be given to illustrate applications of our new results established. [ABSTRACT FROM AUTHOR]

This work is devoted to stochastic functional differential equations (SFDEs) with infinite delay. First, existence and uniqueness of the solutions of such equations are examined. Because the solutions of the delay equations are not Markov, a viable alternative for studying further asymptotic properties is to use solution maps or segment processes. By examining solution maps, this work investigates the Markov properties as well as the strong Markov properties. Also obtained are adaptivity and continuity, mean-square boundedness, and convergence of solution maps from different initial data. This paper then examines the ergodicity of underlying processes and establishes existence of the invariant measure for SFDEs with infinite delay under suitable conditions. [ABSTRACT FROM AUTHOR]

Abstract: This paper establishes the stochastic LaSalle theorem to locate limit sets for stochastic functional differential equations with infinite delay, from which some criteria on attraction, boundedness, stability and robustness are obtained. To illustrate the applications of our results clearly, this paper considers a scalar stochastic integro-differential equation with infinite delay as an example. [Copyright &y& Elsevier]

This article considers a class of nonlocal stochastic functional differential equations with infinite delay whose coefficients are dependent the pth moment and establishes the existence-and-uniqueness theorem under the conditions that are similar to the classical linear growth condition and the Lipschitz condition. Compared with the existing results, the conditions of this article are easier to test. [ABSTRACT FROM AUTHOR]

Abstract: Without the linear growth condition or the one-sided linear growth condition, this paper discusses whether or not stochastic noise feedback can stabilize a given unstable nonlinear functional system . Since may defy the linear growth condition or the one-sided linear growth condition, this system may explode in a finite time. To stabilize this system, this paper stochastically perturbs this system into the stochastic functional differential system by two independent Brownian motions and . This paper shows that the Brownian motion may suppress the potential explosion of the solution for appropriate . Moreover, for sufficiently large , this stochastic functional system is exponentially stable. These results can be used to examine stochastic stabilization. [ABSTRACT FROM AUTHOR]

This paper establishes the existence-and-uniqueness theorems of the global solution to a class of stochastic delay differential equations only with the local Lipschitz condition but does not consider the linear growth condition and studies stochastic ultimate boundedness and the moment average in time. Here we allow the coefficients of these equations to be polynomial or their growths are controlled by polynomial speed. To illustrate our idea clearly, we consider a scalar equation and a special n-dimensional equation carefully. [ABSTRACT FROM AUTHOR]

Abstract: We regard the stochastic functional differential equation with infinite delay as the result of the effects of stochastic perturbation to the deterministic functional differential equation , where is defined by . We assume that the deterministic system with infinite delay is exponentially stable. In this paper, we shall characterize how much the stochastic perturbation can bear such that the corresponding stochastic functional differential system still remains exponentially stable. [Copyright &y& Elsevier]

In general, population systems are often subject to environmental noise. To examine whether the presence of such noise affects population systems significantly, we stochastically perturb the delay Kolmogorov-type system [image omitted] = diag(x1(t),..., xn(t))f(x(t), x(t - δ(t))) into the stochastic delay differential equation dx(t) = diag(x1(t),..., xn(t))[f(x(t), x(t - δ(t)))dt + g(x(t), x(t - δ(t)))dw(t)]. Under the traditional diagonal dominance condition, we study the existence and uniqueness of the global positive solution of this stochastic system, and its asymptotic-bound properties. These properties are natural requirements from the biological point of view. As the special cases, we also discuss some stochastic Lotka-Volterra systems. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we stochastically perturb the functional Kolmogorov-type system into the stochastic functional differential equation This paper studies existence and uniqueness of the global positive solution, and its asymptotic bound properties and moment average in time. These properties are natural requirements from the biological point of view. As the special cases, we discuss the various stochastic Lotka–Volterra systems. [Copyright &y& Elsevier]

In this paper we investigate the stability in distribution for a class of stochastic functional differential equations (SFDEs), which include stochastic differential delay equations (SDDEs). Although stability in distribution has been studied by several authors recently, there is so far no stability-in-distribution criterion on SFDEs where the terms involved the delay components are highly nonlinear (not bounded by linear functions). In this paper we will establish the sufficient criteria on the stability in distribution for a class of highly nonlinear SFDEs. Two examples will be given to illustrate our new results. We also explain the reason why the existing stability-in-distribution criteria are not applicable by these two examples. [ABSTRACT FROM AUTHOR]

This paper is concerned with approximations of invariant probability measures for stochastic functional differential equations (SFDEs) using a backward Euler-Maruyama (BEM) scheme under one-sided Lipschitz condition on the drift coefficient. Firstly, the strong convergence of the numerical "segment sequence" from the BEM scheme on finite time interval [ 0 , T ] is established. In addition, it is also demonstrated that the numerical segment sequence from the BEM scheme is a Markov process, and the corresponding discrete-time semigroup generated by this BEM scheme admits a unique numerical invariant probability measure. Finally, it is revealed that the numerical invariant probability measure converges to the underlying one in a Wasserstein distance. [ABSTRACT FROM AUTHOR]

This paper considers a class of neutral-type stochastic functional differential equations with infinite delay (IDNSFDEs) and highly nonlinear coefficients (i.e., coefficients do not satisfy the linear growth condition). These systems are often unstable. Main aim of this paper is to design a delay feedback control to make them become stable with general decay rate. This general decay stability contains the exponential stability and the polynomial stability. Finally, to illustrate our results more clearly, as examples, this paper also introduces unstable scalar neutral-type stochastic integro-differential equations and discusses their exponential and polynomial stabilisation by delay feedback controls, respectively. [ABSTRACT FROM AUTHOR]