The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion

Abstract

Abstract. In this paper we investigate the existence, uniqueness and exponential asymptotic behavior of mild solutions to stochastic delay evolution equations perturbed by a fractional Brownian motion B H Q (t):with Hurst parameter H ∈ (1/2, 1). We also consider the existence of weak solutions.

Stochastic delay fractional evolution equations driven by fractional Brownian motion

“…Therefore, the existence result of mild solutions in this paper generalizes the corresponding existence result in [15].…”

Controllability of neutral stochastic functional integro-differential equations driven by fractional Brownian motion

“…Now, we end this subsection by stating the following result which is fundamental to prove our result. It can be proved by similar arguments as those used to prove Lemma 2 in [4].…”

“…Many studies of the solutions of stochastic equations in an infinite dimensional space with a fBm have been emerged recently, see [1,2,3,4,6,11,13]. The literature related to neutral stochastic partial functional integro-differential equations driven by a fBm is not vast.…”

Existence and stability results for neutral stochastic delay differential equations driven by a fractional Brownian motion

“…Maslowski and Nualart [12] have studied the existence and uniqueness of a mild solution for nonlinear stochastic evolution equations in a Hilbert space driven by a cylindrical fBm under some regularity and boundedness conditions on the coefficients. Recently, Caraballo and et al [13] investigated the existence and uniqueness of mild solutions to stochastic delay equations driven by fBm with Hurst parameter . An existence and uniqueness result of mild solutions for a class of neutral stochastic differential equation with finite delay, driven by an fBm in a Hilbert space has been investigated [14] in Boufoussi and Hajji.…”

“…An existence and uniqueness result of mild solutions for a class of neutral stochastic differential equation with finite delay, driven by an fBm in a Hilbert space has been investigated [14] in Boufoussi and Hajji. The asymptotic behavior of solutions for stochastic differential equations with fBm has only been investigated by a few authors [13], [14], [15]. Moreover Nguyen [16] has studied the asymptotic behaviors of mild solutions to neutral stochastic differential equations driven by an fBm.…”