Limiting behavior of invariant or periodic measure of Hopfield neural models driven by locally Lipschitz Lévy noise

This paper is concerned with the existence and limiting behavior of invariant probability measures or periodic probability measures for a type of widely used Hopfield-type lattice models with two nonlinear terms of arbitrary polynomial growth on the entire integer set $ \mathbb{Z}^d $ driven by nonlinear white noise and Lévy noise. First, when the noise intensity is within a controllable range, we prove that the family probability distribution laws solutions and use the weak convergence method to prove the existence of invariant probability measures. Then, when the terms that change over time are periodic we also discussed the periodic probability measures existence in a weighted $ \ell_\rho^2 $ space. Finally, the limiting behavior of the collection of all invariant or periodic probability measures weakly compact are studied for Hopfield models driven by nonlinear white noise and Lévy noise about with noise intensity.