Recurrent Measures, Exponential Mixing and Limit Theorems for Stochastic Evolution Equations with Delay and Monotone Coefficients

Abstract

We investigate a class of stochastic evolution equations with monotone coefficients and finite delay, motivated by non-equilibrium physical systems in which both memory effects and stochastic perturbations play essential roles. The inherent non-Markovian nature induced by delay challenges the application of classical ergodic theory. To overcome this challenge, we introduce the concept of recurrent entrance measures, which extends the classical notion of invariant measures to non-autonomous settings. By generalizing Shcherbakov’s comparability principle to a stochastic variational framework, we prove that these measures inherit the full spectrum of recurrence properties (e.g., stationary, periodic, quasi-periodic, almost periodic, Levitan almost periodic, Bohr almost automorphic, pseudo-periodic, Birkhoff recurrent, pseudo-recurrent and Poisson stable) from the system’s coefficients, marking a shift from pathwise to statistical descriptions of recurrence. For autonomous systems, we prove uniform exponential mixing in the Wasserstein metric and further establish strong laws of large numbers and central limit theorems for solution maps. Our results provide a unified statistical framework for stochastic evolution systems with delay, with applications to physical models exhibiting memory and stochastic forcing.