Stochastic Lattice Dynamical Systems with Fractional Noise

Abstract

This article is devoted to study stochastic lattice dynamical systems driven by a fractional Brownian motion with Hurst parameter H ∈ (1/2, 1). First of all, we investigate the existence and uniqueness of pathwise mild solutions to such systems by the Young integration setting and prove that the solution generates a random dynamical system. Further, we analyze the exponential stability of the trivial solution.

Pathwise solution to rough stochastic lattice dynamical system driven by fractional noise

“…Following the same procedure, one can obtain (6g). In particular, other desired estimates were proved in [4,Lemma 3.2]. Therefore, the proof of this Lemma is completed.…”

“…for all 0 ≤ s ≤ t. The first one can be obtained by the energy inequality and the last two follow by the mean value theorem, see also [4]. Furthermore, using these inequalities we can deduce the following properties of S(t).…”

“…Later on, the existence of a global forward attracting set of a stochastic lattice system with a Caputo fractional time derivative was established in the weak mean-square topology [32]. Recently, stochastic lattice dynamical systems driven by fractional Brownian motion with H ∈ (1/2, 1) was proved to have a unique pathwise mild solution, which is exponentially stable under suitable conditions [4]. There has, however, been little mention about the SLDS in the rough paths case, say, H ∈ (0, 1/2].…”

Analysis of a stochastic SIR model with fractional Brownian motion

“…In this section we recall some notations and necessary concepts for our objectives (see [3]). Let us start by considering two given values T 1 < T 2 .…”

Periodic solutions in distribution of stochastic lattice differential equations

“…Therefore, as models, SLDEs have attracted extensive attention. For instance, refer to [2,3,5,6,7,19,26,30,37,38,39,42]. As far as we know, there are few studies about periodic solutions of SLDEs.…”