Uniform-in-Time Weak and Ergodic Error Estimates of a Nonlinearity-Explicit Full Discretization for Superlinear SPDEs Driven by Multiplicative Noise

2026/07/21 by Jingjing Cai, Zhihui Liu, Xiaoming Wu
Computer Science · Mathematics · #cs.NA #math.NA

paper · pdf

arxiv created 2026/07/29 · arxiv updated 2026/07/30

Abstract

For a class of superlinear SPDEs driven by multiplicative noise, we prove an (essentially) sharp uniform-in-time (UIT) weak convergence rate for the nonlinearity-explicit Galerkin tamed Euler method (GTEM). Under standard monotonicity assumptions, the proof combines Malliavin calculus with regularity theory for the associated backward Kolmogorov equation (BKE), leading to UIT moment, Hölder, and Malliavin estimates, along with regularity estimates for the BKE solution. These estimates, together with a weak error decomposition and Malliavin integration by parts (IBP) formula, then yield a UIT weak convergence rate τ^ρ+λ_N^-(ρ+γ/2) for any ρ∈ (0,1), where γ∈[0,1) quantifies the assumed spatial Sobolev regularity. Consequently, we obtain a sharp ergodic error estimate between the exact and numerical invariant measures. Numerical experiments support the theory.

Citations

• Strong and Weak Convergence Rates of a Spatial Approximation for Stochastic Partial Differential Equation with One-sided Lipschitz Coefficient

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