Numerical Unique Ergodicity of Monotone SDEs driven by Nondegenerate Multiplicative Noise
arXiv:2401.01112 · doi:10.1007/s10915-025-02902-4
Abstract
We first establish the unique ergodicity of the stochastic theta method (STM) with for monotone SODEs, without growth restriction on the coefficients, driven by nondegenerate multiplicative noise. The main ingredient of the arguments lies in constructing new Lyapunov functions involving the coefficients, the stepsize, and and deriving a minorization condition for the STM. We then generalize the arguments to the Galerkin-based full discretizations for 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 Galerkin-based full discretizations are uniquely ergodic for any interface thickness. Numerical experiments verify our theoretical results.
arXiv comment21 pages, 2 figures; to appear in J. Sci. Comput
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