非退化乘性噪声驱动的单调SDE的数值唯一遍历性
Numerical Unique Ergodicity of Monotone SDEs driven by Nondegenerate Multiplicative Noise
AI总结：
本文证明了非退化乘性噪声驱动、无系数增长限制的单调SODE的STM及单调SPDE的Galerkin全离散化的唯一遍历性，并应用于随机Allen--Cahn方程，通过数值实验验证了理论。
摘要：
我们首先证明了由非退化乘性噪声驱动、对系数无增长限制的单调SODE的随机theta方法（STM）（$θ\in [1/2, 1]$）的唯一遍历性。论证的关键在于构造涉及系数、步长和$θ$的新Lyapunov函数，并推导STM的minorize条件。随后，我们将该论证推广至一类由无穷维非退化乘性迹类噪声驱动的单调SPDE的基于Galerkin全离散化。将这些结果应用于随机Allen--Cahn方程表明，其基于Galerkin的全离散化对任意界面厚度均具有唯一遍历性。数值实验验证了我们的理论结果。
英文摘要：
We first establish the unique ergodicity of the stochastic theta method (STM) with $θ\in [1/2, 1]$ 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 $θ$ 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.
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