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<p>This paper investigates exponential mixing and limiting behaviors of invariant measures for stochastic delay lattice systems under small noise perturbations. While the classical Krylov–Bogolyubov method successfully demonstrates the existence of invariant measures, the stronger assertions regarding uniqueness and exponential mixing require alternative approaches. Departing from this classical framework, we simultaneously establish ergodicity and exponential mixing of invariant measures by utilizing the contraction properties of the Markov transition semigroup in terms of the 1-order Wasserstein distance. Another pivotal aspect of this paper is our exploration of the limiting behaviors of invariant measures. We rigorously demonstrate that, as at least one of the noise intensity or time delay approaches zero, the invariant measure of the stochastic delay lattice system with small noise converges to that of the corresponding limiting system. Furthermore, we establish quantitative convergence rates for the limiting process.</p>

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Exponential mixing and limiting behaviors of invariant measures of stochastic delay lattice systems with small noise

  • 期刊论文

  • 发表日期：

    2026年3月14日

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  • Sa Xiao,
  • Jun Shen

摘要

This paper investigates exponential mixing and limiting behaviors of invariant measures for stochastic delay lattice systems under small noise perturbations. While the classical Krylov–Bogolyubov method successfully demonstrates the existence of invariant measures, the stronger assertions regarding uniqueness and exponential mixing require alternative approaches. Departing from this classical framework, we simultaneously establish ergodicity and exponential mixing of invariant measures by utilizing the contraction properties of the Markov transition semigroup in terms of the 1-order Wasserstein distance. Another pivotal aspect of this paper is our exploration of the limiting behaviors of invariant measures. We rigorously demonstrate that, as at least one of the noise intensity or time delay approaches zero, the invariant measure of the stochastic delay lattice system with small noise converges to that of the corresponding limiting system. Furthermore, we establish quantitative convergence rates for the limiting process.