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

Abstract

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 $Z^d$ 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 ${\xi }_{i,j}$ between the $i$th and $j$th 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 ${\ell }_{\varrho }^2$. 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 $L^2(\tilde{\Omega },{\ell }_{\varrho }^2)$ over a complete filtered probability space $(\tilde{\Omega },\tilde{F},{\left\{{\tilde{F}}_t\right\}}_{t\in R},P)$. 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.