Attractors for Stochastic Lattice Dynamical Systems

Abstract

We consider a one-dimensional lattice with diffusive nearest neighbor interaction, a dissipative nonlinear reaction term and additive independent white noise at each node. We prove the existence of a compact global random attractor within the set of tempered random bounded sets. An interesting feature of this is that, even though the spatial domain is unbounded and the solution operator is not smoothing or compact, pulled back bounded sets of initial data converge under the forward flow to a random compact invariant set.

Random attractors for stochastic lattice systems with non-Lipschitz nonlinearity

“…We prove the existence of a random global pullback attractor for such systems, extending in this way the results given in [3] (see [11] for the case of a multiplicative noise, and also [21] for a partial dissipative stochastic lattice dynamical system with additive noise). The main difference with [3] is the fact that we do not assume conditions ensuring the uniqueness of solutions for the Cauchy problem. Hence, we use the theory of multi-valued random dynamical systems [6] in order to prove the existence of the pullback attractor.…”

“…Hence, we use the theory of multi-valued random dynamical systems [6] in order to prove the existence of the pullback attractor. Comparing our results with the single-valued case from [3], the main technical difficulty appears in proving the measurability of the attractor.…”

Attractors for stochastic lattice dynamical systems with a multiplicative noise

“…27 in [26] (see [5] for the proof). Although it may be possible to apply a more general result from Caraballo et al [10], the following one will fit our purposes.…”

“…This may be the result of an environmental effect on the whole domain of the system. The system with an additive noise was studied in Bates et al [5].…”

A Random Model for Immune Response to Virus in Fluctuating Environments

“…Proposition 1. [5,9,10] Let B ∈ D(X) be an absorbing set for the continuous random dynamical system {S(t, ω)} t≥0,ω∈Ω which is closed and satisfies the asymptotic compactness condition for a.e. ω ∈ Ω , i.e., each sequence x n ∈ S(t n , θ −t n , B(θ −t n ω)) has a convergent subsequence in X when t n → ∞.…”