Asymptotic description of stochastic neural networks. II - Characterization of the limit law

TL;DR

This paper advances the asymptotic analysis of stochastic neural networks by characterizing the limit law using Large Deviation Principles, identifying a unique stationary measure, and providing methods for its computation.

Contribution

It introduces a novel characterization of the limit law for stochastic neural networks via a unique stationary measure and offers practical computation techniques.

Findings

Existence of a unique minimum of the rate function H.

The second marginal of the measure is a stationary Gaussian.

Convergence results established using Large Deviation Principles.

Abstract

We continue the development, started in of the asymptotic description of certain stochastic neural networks. We use the Large Deviation Principle (LDP) and the good rate function H announced there to prove that H has a unique minimum mu_e, a stationary measure on the set of trajectories. We characterize this measure by its two marginals, at time 0, and from time 1 to T. The second marginal is a stationary Gaussian measure. With an eye on applications, we show that its mean and covariance operator can be inductively computed. Finally we use the LDP to establish various convergence results, averaged and quenched.