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The polynomial stability problem of stochastic delay differential equations has been studied in recent years. In contrast, there are relatively few works on stochastic partial differential equations with pantograph delay. The present paper is devoted to investigating large-time asymptotic properties of solutions for stochastic pantograph delay evolution equations with nonlinear multiplicative noise. We first show that the mild solutions of stochastic pantograph delay evolution equations with nonlinear multiplicative noise tend to zero with general decay rate (including both polynomial and logarithmic rates) in the p th moment and almost sure senses. The analysis is based on the Banach fixed point theorem and various estimates involving the gamma function. Moreover, by using a generalized version of the factorization formula and exploiting an approximation technique and a convergence analysis, we construct the nontrivial equilibrium solution, defined for t ∈ ℝ , for stochastic pantograph delay evolution equations with nonlinear multiplicative noise. In particular, the uniqueness, Hölder regularity in time and general stability, in the p th moment and almost sure senses, of the nontrivial equilibrium solution are established. [ABSTRACT FROM AUTHOR]

In this paper, we prove a result of existence and uniqueness of solutions to coupled forward–backward stochastic differential equations with delayed generators under a Lipschitz condition. [ABSTRACT FROM AUTHOR]

We consider stochastic 2D-Stokes equations with unbounded delay in fractional power spaces and moments of order p ≥ 2 driven by a tempered fractional Brownian motion (TFBM) B σ , λ (t) with − 1 / 2 < σ < 0 and λ > 0. First, the global existence and uniqueness of mild solutions are established by using a new technical lemma for stochastic integrals with respect to TFBM in the sense of p th moment. Moreover, based on the relations between the stochastic integrals with respect to TFBM and fractional Brownian motion, we show the continuity of mild solutions in the case of λ → 0 , σ ∈ (− 1 / 2 , 0) or λ > 0 , σ → σ 0 ∈ (− 1 / 2 , 0). In particular, we obtain p th moment Hölder regularity in time and p th polynomial stability of mild solutions. This paper can be regarded as a first step to study the challenging model: stochastic 2D-Navier–Stokes equations with unbounded delay driven by tempered fractional Gaussian noise. [ABSTRACT FROM AUTHOR]

In this paper, we establish an averaging principle for neutral stochastic fractional differential equations with non-Lipschitz coefficients and with variable delays, driven by Lévy noise. Our result shows that the solutions of the equations concerned can be approximated by the solutions of averaged neutral stochastic fractional differential equations in the sense of convergence in mean square. As an application, we present an example with numerical simulations to explore the established averaging principle. [ABSTRACT FROM AUTHOR]

Main goal of this paper is to formulate possibly simple and easy to verify criteria on existence of the unique attracting probability measure for stochastic process induced by generalized iterated function systems with probabilities (GIFSPs). To do this, we study the long-time behavior of trajectories of Markov-type operators acting on product of spaces of Borel measures on arbitrary Polish space. Precisely, we get the desired geometric rate of convergence of sequences of measures under the action of such operator to the unique distribution in the Hutchinson–Wasserstein distance. We apply the obtained results to study limiting behavior of random trajectories of GIFSPs as well as stochastic difference equations with multiple delays. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of almost periodic solutions for fractional impulsive neutral stochastic differential equations with infinite delay in Hilbert space. The main conclusion is obtained by using fractional calculus, operator semigroup and fixed point theorem. In the end, we give an example to illustrate our main results. [ABSTRACT FROM AUTHOR]

In this work we study the existence, uniqueness and asymptotic behavior of mild solutions for neutral stochastic partial integrodifferential equations with infinite delays. To prove the results, we use the theory of resolvent operators as developed by R. Grimmer [13], as well as a version of the fixed point principle. We establish sufficient conditions ensuring that the mild solutions are exponentially stable in th-moment. An example is provided to illustrate the abstract results. [ABSTRACT FROM AUTHOR]

In this note we prove that a fractional stochastic delay differential equation which satisfies natural regularity conditions generates a continuous random dynamical system on a subspace of a Hölder space which is separable. [ABSTRACT FROM AUTHOR]