Hopfield neural lattice models with locally Lipschitz coefficients driven by Lévy noise

In this article, we study the global-in-time solvability and long-term dynamics of a wide class of infinite-dimensional Hopfield neural models on Zd of infinitely many ODEs with a family of locally Lipschitz coefficients driven by Lévy noise. There are three new features of this stochastic model: (1)The Lévy noise is characterized by two sequence of mutually independent two-sided (including negative initial times) Wiener processes and Poisson random measures; (2)The diffusion coefficients of the Lévy noise are locally Lipschitz associated with an appropriate weight; (3)The connection strength ξi,j between the ith and jth neurons has a finite reciprocal-weighted aggregate efficacy in a weak sense. This Lévy noise driven lattice equation is formulated as an abstract one in an infinite-dimensional weighted Hilbert space ℓϱ2. Both global-in-time well-posedness and long-time dynamics of this abstract stochastic system are investigated under certain conditions. In particular, we show that the long-time dynamics of the stochastic systems can be captured by a weakly compact and weakly attracting mean random attractor in the Bochner space L2(Ω̃,ℓϱ2) over a complete filtered probability space (Ω̃,F̃,{F̃t}t∈R,P). It seems that this is the first time to study the well-posedness and dynamics of lattice Hopfield neural models with locally Lipschitz coefficients driven by Lévy noise even in the autonomous case.