Invariant and periodic measures in classical spin systems on infinite lattices with highly degenerate noise

TL;DR

This paper studies infinite-dimensional classical spin systems with degenerate noise, proving existence, uniqueness, and ergodic properties of invariant and periodic measures under weak dissipation and local interactions.

Contribution

It establishes the existence and uniqueness of invariant and periodic measures for infinite spin systems with highly degenerate noise, including ergodic and convergence results.

Findings

Unique invariant and periodic measures exist under certain conditions.

Geometric ergodicity holds for finite-volume Markov systems.

Weak convergence of measures yields invariant and periodic measures for the infinite system.

Abstract

In this paper, we consider the classical spin systems on unbounded lattices given by infinite-dimensional stochastic differential equations (SDEs). We assume that the stochastic forcing acts only on one particle. The other particles are not subject to stochastic forcing directly, but interact with their nearest neighbouring particles. Under the above highly degenerate noise setting, with some mild assumptions on the local interaction of each particle such as weak dissipation, we obtain the existence, uniqueness and the Markovian property of weak martingale solutions. We prove that the one-dimensional noise can propagate to any spin particle in the system in the sense that there exists a unique invariant/periodic measure and geometric ergodicity holds for the Markovian system when restricted to any finite volume. We then prove the finite-dimensional invariant measure and the average of lifted periodic measure are tight, and weak convergent subsequence gives an invariant and periodic measures of the infinite spin systems, respectively, in the time-homogeneous or time-periodic cases.