Lyapunov function for interacting reinforced stochastic processes via Hopfield’s energy function

Abstract

In 1984, Hopfield introduced an artificial neural network to understand the memory in living organisms. Under a condition of symmetry, he showed that there exists a Lyapunov function (known as energy function) for the network. In 2015, Budhiraja, Dupuis and Fischer introduced an idea to construct Lyapunov functions for nonlinear Markov processes via relative entropy. In this article, we introduce an approach based on the energy function of Hopfield networks to obtain Lyapunov functions for a class of interacting reinforced stochastic processes. Our result is an alternative to Budhiraja, Dupuis and Fischer’s approach and it works for the case of processes with finitely many 2-dimensional probability measures. We also bring from the neural network framework results about the total stability of differential equations. Finally, we include an application of the results to the study of differential equations associated to n≥2 vertex reinforced random walks on two-vertex graphs.