Optimal Weak Order and Approximation of the Invariant Measure with a Fully-Discrete Euler Scheme for Semilinear Stochastic Parabolic Equations with Additive Noise

Qiu Lin and Ruisheng Qi ()
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Qiu Lin: School of Mathematics and Statistics, Yancheng Teachers University, Yancheng 224002, China
Ruisheng Qi: School of Mathematics and Statistics, Yancheng Teachers University, Yancheng 224002, China

Mathematics, 2023, vol. 12, issue 1, 1-29

Abstract: In this paper, we consider the ergodic semilinear stochastic partial differential equation driven by additive noise and the long-time behavior of its full discretization realized by a spectral Galerkin method in spatial direction and an Euler scheme in the temporal direction, which admits a unique invariant probability measure. Under the condition that the nonlinearity is once differentiable, the optimal convergence orders of the numerical invariant measures are obtained based on the time-independent weak error, but not relying on the associated Kolmogorov equation. More precisely, the obtained convergence orders are O ( λ N − γ ) in space and O ( τ γ ) in time, where γ ∈ ( 0 , 1 ] from the assumption ∥ A γ − 1 2 Q 1 2 ∥ L 2 is used to characterize the spatial correlation of the noise process. Finally, numerical examples confirm the theoretical findings.

Keywords: stochastic partial differential equation; invariant measure; ergodicity; weak approximation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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