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Based on the Girsanov theorem for a kind of Volterra-Gaussian process, which are the generalization of fractional Brownian motion, Liouville fractional Brownian motion, and fractional Ornstein-Uhlenbeck process, we establish the Harnack inequalities for a class of stochastic functional differential equations driven by a kind of Volterra-Gaussian processes with a subordinator by an approximation technique. Some known results have been generalized and improved. [ABSTRACT FROM AUTHOR]

This manuscript focuses on a class of stochastic functional differential equations driven by time-changed Brownian motion. By utilizing the Lyapunov method, we capture some sufficient conditions to ensure that the solution for the considered equation is η-stable in the pth moment sense. Subsequently, we present some new criteria of the η-stability in mean square by using time-changed Itô formula and proof by contradiction. Finally, we provide some examples to demonstrate the effectiveness of our main results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a class of stochastic differential equations driven by the time-changed Brownian motion and impulsive stochastic differential equations driven by the time-changed Brownian motion. By establishing some time-changed retarded Gronwall-like inequalities and using the Girsanov transformation argument, we establish the quadratic transportation inequalities for the law of the solution of those equations. [ABSTRACT FROM AUTHOR]

In this paper, we focus on a class of stochastic differential equations driven by the standard Brownian motion and fractional Brownian motion with Hurst parameter 1 / 2 < H < 1 under a weaker conditions than Lipschitz one. In the sense of the pathwise Riemann–Stieltjes integral, we prove the convergence of solutions for the considered equations. By making use of some transformation techniques and approximation means, we obtain some sufficient conditions on the viability for the stochastic systems under investigation. Subsequently, by using some distance functions, we establish some necessary conditions and some sufficient conditions for the viability property with respect to a given non-empty closed set K, but K can be not smooth. As applications of the results, we explore the comparison theorems about the mixed stochastic differential equations driven by fractional Brownian motion under some non-Lipschitz conditions. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with a class of coupled multidimensional stochastic differential equations driven by fractional Brownian motion with Hurst parameter H ∈ (1 / 2 , 1) . Using pathwise approach and based on Perov's fixed point theorem, we prove that the existence and uniqueness of solution for the equations considered under some local Lipschitz conditions. Subsequently, by establishing some priori estimates, we obtain a viability result for the stochastic systems under investigation. The sufficient and necessary condition is also an alternative global existence result for the fractional differential equations with restrictions on the state. Finally, by direct and inverse images for fractional stochastic tangent sets, we establish the deterministic necessary and sufficient conditions which control that the solution for the coupled stochastic systems under investigation evolves in some particular sets. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with a class of coupled neutral stochastic partial differential equations driven by fractional Brownian motion with Hurst parameter H ∈ (0 , 1 / 2) ∪ (1 / 2 , 1). On account of the Perov's fixed point theorem and semigroup theory, we prove the existence and uniqueness of mild solution. Subsequently, by using delay integral inequalities, we identify the global attracting sets of the equations under investigation. Furthermore, we obtain some sufficient conditions that ensure the exponential decay of mild solutions in the pth moment. Lastly, we present an example to illustrate our theory in this work. [ABSTRACT FROM AUTHOR]

In this paper, we consider the existence and uniqueness of the mild solution for a class of coupled fractional stochastic evolution equations driven by the fractional Brownian motion with the Hurst parameter H ∈ 1 / 4 , 1 / 2 . Our approach is based on Perov's fixed-point theorem. Furthermore, we establish the transportation inequalities, with respect to the uniform distance, for the law of the mild solution. [ABSTRACT FROM AUTHOR]

In this paper, by using a transformation formula for fractional Brownian motion (fBm), we prove the existence of weak solutions to stochastic differential equations driven by an additive fBm with Hurst parameter 1 / 2 < H < 1 under the linear growth condition. Furthermore, we also consider the uniqueness in law and the pathwise uniqueness of the weak solution. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of stochastic fractional evolution equations with infinite delay and a fractional Brownian motion in a Hilbert space. By the stochastic analysis technique, we establish the existence and uniqueness of mild solutions for these equations under non-Lipschitz condition with Lipschitz conditions being considered as a special case. An example is provided to illustrate the theory. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with a class of second-order neutral stochastic functional differential equations driven by a fractional Brownian motion with Hurst parameter $1/2<\hbar <1$ on the Hilbert space. By combining some stochastic analysis theory and new integral inequality techniques, we identify the global attracting sets of the equations under investigation. Some sufficient conditions ensuring the exponential decay of mild solutions in the pth moment to the stochastic systems are obtained. Last, an example is presented to illustrate our theory in the work. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of stochastic functional differential equations driven by the time-changed Brownian motion. Some sufficient conditions ensuring the existence of the global attracting sets of considered equations are presented. As applications, some new criteria for the exponential stability in mean square of the considered equations are obtained. Lastly, some examples are investigated to illustrate the theory. [ABSTRACT FROM AUTHOR]

In this paper, we prove a viability result for stochastic functional differential equation with infinite memory driven by a fractional Brownian motion with Hurst parameter 1 ∕ 2 < H < 1 , using pathwise approach. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of stochastic differential equations with a time-dependent diffusion driven by a fractional Brownian motion with Hurst parameter 1 ∕ 2 < H < 1. By using a transformation formula for fractional Brownian motion, we prove the existence of weak solutions to this kind of equations under the linear growth condition, but the drift can be discontinuous. Some known results are improved. • We prove the existence and uniqueness of a strong solutions for (5) when b 1 is linear growth but is discontinuous in x. • Our results improve the results of the Theorem 3.2 and 3.4 in [1]. • Using the unique strong solution for (5), we obtain existence of the weak solution for (1) only assuming that satisfies (3) without any continuity assumption. [ABSTRACT FROM AUTHOR]