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ABSTRACT

We establish the unique ergodicity of the Markov chain generated by the stochastic theta method (STM) with \theta \in [1/2, 1] for monotone SODEs and SPDEs driven by multiplicative noise. The main ingredient of the arguments lies in constructing new Lyapunov functions involving the coefficients, the stepsize, and \theta, and the irreducibility and the strong Feller property for the STM. We also generalize the arguments to a class of monotone SPDEs driven by infinite-dimensional nondegenerate multiplicative trace-class noise. Applying these results to the stochastic Allen-Cahn equation indicates that its drift-implicit Euler scheme is uniquely ergodic for any interface thickness.