A note on the generation of random dynamical systems from fractional stochastic delay differential equations

Abstract

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.

Young differential delay equations driven by Hölder continuous paths

“…for some constant 1 > δ > 1−ν ν . Assumption (4) is weaker than the global Lipschitz continuity of Dg, as seen in [1], [6] or [19]. Furthermore, we show that the solution is differentiable with respect to the initial function η and give an estimate for the growth of the solution.…”

“…[4], [5], [9], [10], [11], [16], [17], [20],... and the references therein). For studies on delay equations, we refer to [1], [2], [3], [6].…”

Stochastic shell models driven by a multiplicative fractional Brownian-motion

“…is a separable space since V is itself separable (see [13], [14] and [11]). It is easy to see that C γ ([0, T ]; V ) ⊂ C 0,β ′ ([0, T ]; V ), and therefore this latter space is the one that we should take when considering the fBm.…”

Pullback attractors for stochastic Young differential delay equations

“…Notice that the same question for non-delay Young differential equations is well-studied in [19], [12], [11], [13], where one can prove that the system generates a random dynamical system which possesses a random attractor. For the delay system (1.1), the existence and uniqueness of the solution and the generation of a random dynamical system is affirmed in [14] and [10], but the question on asymptotic stability is still open. Our aim in this paper is to show that under the assumptions H 1 , H 2 , H 3 , the system (1.1) will generate a random dynamical system by means of its solution flow, and furthermore it possesses a random pullback attractor if the nonlinear term and stochastic term are small.…”