Ergodicity for stochastic neural field equations

TL;DR

This paper studies the long-term behavior of stochastic neural field models, establishing conditions for ergodicity, mixing, and the uniqueness of invariant measures in unbounded domains with Gaussian noise.

Contribution

It provides new conditions ensuring exponential ergodicity and mixing for neural field equations with noise, extending understanding of their long-term dynamics.

Findings

Existence of invariant probability measures under certain conditions.

Exponential ergodicity and mixing established for the model.

Uniqueness of invariant measure with finite second moments proved.

Abstract

We investigate the well-posedness and long-time behavior of a general continuum neural field model with Gaussian noise on possibly unbounded domains. In particular, we give conditions for the existence of invariant probability measures by restricting the solution flow to an invariant subspace with a nonlocal metric. Under the assumption of a sufficiently large decay parameter relative to the noise intensity, the growth of the connectivity kernel, and the Lipschitz regularity of the activation function, we establish exponential ergodicity and exponential mixing of the associated Markovian Feller semigroup and the uniqueness of the invariant measure with second moments.