由乘性噪声驱动的超线性随机偏微分方程的非线性显式全离散化的时间一致弱收敛和遍历误差估计
Uniform-in-Time Weak and Ergodic Error Estimates of a Nonlinearity-Explicit Full Discretization for Superlinear SPDEs Driven by Multiplicative Noise
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中文总结 AI 辅助
研究由乘性噪声驱动的超线性随机偏微分方程，用非线性显式伽辽金驯服欧拉方法，结合马利瓦因微积分与BKE正则性理论，证明时间一致弱收敛速率，得到精确与数值不变测度间遍历误差估计，数值实验支持理论。
中文摘要 AI 辅助
对于一类由乘性噪声驱动的超线性随机偏微分方程，我们证明了非线性显式伽辽金驯服欧拉方法（GTEM）的（本质上）尖锐的时间一致（UIT）弱收敛速率。在标准单调性假设下，证明将马利瓦因微积分与相关反向柯尔莫哥洛夫方程（BKE）的正则性理论相结合，得到UIT矩、赫尔德和马利瓦因估计，以及BKE解的正则性估计。这些估计与弱误差分解和马利瓦因分部积分（IBP）公式一起，对于任何\(\rho\in(0,1)\)，产生UIT弱收敛速率\(\tau^\rho+\lambda_N^{-\rho}\)。因此，我们得到了精确不变测度和数值不变测度之间的尖锐遍历误差估计。数值实验支持该理论。
英文摘要
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 $ρ\in (0,1)$, where $γ\in[0,1)$ 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.
