Stochastic functional partial differential equations with monotone coefficients: Poisson stability measures, exponential mixing and limit theorems

This paper examines Poisson stable (including stationary, periodic, almost periodic, Levitan almost periodic, Bohr almost automorphic, pseudo-periodic, Birkhoff recurrent, pseudo-recurrent, etc.) measures and limit theorems for stochastic functional partial differential equations(SFPDEs) with monotone coefficients. We first show the existence and uniqueness of entrance measure $\mu _{t}$ for SFPDEs by dissipative method (or remoting start). Then, with the help of Shcherbakov's comparability method in character of recurrence, we prove that the entrance measure inherits the same recurrence of coefficients. Thirdly, we show the tightness of the set of measures $\mu _{t}$. As a result, any sequence of the average of $\{\mu _{t}\}_{t\in\mathbb{R} }$ have the limit point $\mu ^{*}$. Further, we study the uniform exponential mixing of the measure $\mu ^{*}$ in the sense of Wasserstein metric. Fourthly, under uniform exponential mixing and Markov property, we establish the strong law of large numbers, the central limit theorem and estimate the corresponding rates of convergence for solution maps of SFPDEs. Finally, we give applications of stochastic generalized porous media equations with delay to illustrate of our results.