Ergodicity for stochastic neural field equations

TL;DR

This work analyzes ergodicity for a broad class of stochastic neural field equations on possibly unbounded domains. By introducing a nonlocal Hilbert subspace $H_1$ and proving Lipschitz and growth bounds for the drift and noise, the authors establish well-posedness and apply Krylov–Bogoliubov methods to obtain invariant measures. Under a quantitative smallness condition that balances (i) kernel-induced interactions and (ii) noise, they prove the existence, uniqueness, and exponential ergodicity/mixing of an invariant measure, with explicit rates. In a monotone, symmetric-kernel setting, they improve convergence rates and show robust ergodic behavior on the invariant subspace, providing rigorous long-time analysis for both additive and multiplicative noise in neural-field models, including unbounded spatial domains.

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.

Theorems & Definitions (39)

• Example 1.1
• Theorem 1.2
• Definition 2.1
• Example 2.2
• Theorem 2.3
• proof
• Definition 2.4
• Theorem 2.5
• proof
• Example 2.6