Stochastic dynamics of a neural field lattice model with state dependent nonlinear noise

Abstract

The well-posedness and long term dynamics of a stochastic nonautonomous neural field lattice system on vector-valued indices Z d driven by state dependent nonlinear noise are investigated in a weighted space of infinite sequences. First the existence and uniqueness of a mean square solution to the lattice system is established under the assumptions that the nonlinear drift and diffusion terms are component-wise continuously differentiable with weighted equi-locally bounded derivatives. Then the existence and uniqueness of a tempered weak pullback mean random attractor associated with the solution is proved. Finally the existence of invariant measures for the neural field lattice system is obtained by uniform tail-estimates and Krylov-Bogolyubov's method.

Uniform large deviation principles of fractional stochastic reaction-diffusion equations on unbounded domains

“…For stochastic lattice systems, the random attractors have been reported in [5,6,14,15,44,45]. Recently, the invariant measures and periodic measures of stochastic lattice systems have been examined in [21,22,23,52,53,66,67,68] and the references therein. In the present paper, we study the large deviation principle of the stochastic lattice system (1.1)-(1.2).…”

“…The traveling solutions and dynamics of deterministic lattice systems have been investigated in [1,2,3,4,8,20,21,22,26,27,30,32,39,42,47] and the references therein. Recently, the dynamics of random lattice systems has been extensively studied, see, e.g., [5,6,12,13,29] for random attractors; and [17,18,19,35,36,43,45,46] for invariant measures and periodic measures. The goal of this paper is to prove the LDP of invariant measures of system (1.1)-(1.2) as ε Ñ 0, which is stated below.…”

Periodic solutions in distribution for stochastic lattice differential equations

“…This enables SLDEs to better reveal objective phenomena. For this reasing, SLDEs have attracted extensive attention; see [3,4,8,14,15,26].…”

Stochastic 3D Globally Modified Navier–Stokes Equations: Weak Attractors, Invariant Measures and Large Deviations

“…Then, by the uniform estimates of the solutions of (1.1) (see Lemma 3.1) and the theory of mean random dynamical systems in [47], the existence and uniqueness of weak pullback mean random attractors can be obtained (see Theorem 3.1). In addition, we would like to mention some studies about the pathwise random attractors [9,26,45,55] and the weak mean random attractors [48,49,53,54].…”