Weak and Wasserstein convergence of periodic measures of stochastic neural field lattice models with Heaviside ’s operators and locally Lipschitz Lévy noises

Abstract

We study the global-in-time well-posedness and periodic measures for a class neural field lattice models (with Heaviside’s operators) defined on the high-dimensional integer set Zd, and driven by locally Lipschitz Lévy noises. We first formulate the stochastic neural field lattice equations into abstract stochastic systems defined in the infinite-dimensional weighted Hilbert space, and then prove the global-in-time well-posedness of the stochastic systems. When the time-dependent forces are periodic, the existence of periodic measures is obtained by the idea of uniform tail-estimates and the Krylov–Bogolyubov method. The limiting behavior of the periodic measures in the weak convergence sense is discussed as the noises intensities approach zero. A stronger convergence of the periodic measures is also established in the Wasserstein distance sense.