Numerical Unique Ergodicity of Monotone SDEs driven by Nondegenerate Multiplicative Noise

Abstract

We first establish the unique ergodicity of the stochastic theta method (STM) with $\theta \in [1/2,1]$ 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 $\theta$ 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.

21 pages, 2 figures; to appear in J. Sci. Comput

References in corpus (3)

• Rate of Convergence of Space Time Approximations for stochastic evolution equations
• Numerical Ergodicity and Uniform Estimate of Monotone SPDEs Driven by Multiplicative Noise
• Numerical Ergodicity of Stochastic Allen--Cahn Equation driven by Multiplicative White Noise

Cited by in corpus (1)

• Numerical Ergodicity and Uniform Estimate of Monotone SPDEs Driven by Multiplicative Noise