This article investigates the weak approximation towards the invariant measure of semi-linear stochastic differential equations (SDEs) under non-globally Lipschitz coefficients. For this purpose, we propose a linear-theta-projected Euler (LTPE) scheme, which also admits an invariant measure, to handle the potential influence of the linear stiffness. Under certain assumptions, both the SDE and the corresponding LTPE method are shown to converge exponentially to the underlying invariant measures, respectively. Moreover, with time-independent regularity estimates for the corresponding Kolmogorov equation, the weak error between the numerical invariant measure and the original one can be guaranteed with an order one. Numerical experiments are provided to verify our theoretical findings.

翻译：本文研究了非全局Lipschitz系数下半线性随机微分方程(SDEs)不变测度的弱逼近问题。为此，我们提出了一种线性-θ-投影欧拉(LTPE)格式，该格式同样具有不变测度，以处理线性刚度项的潜在影响。在特定假设条件下，本文分别证明了SDE及其对应的LTPE方法指数级收敛于各自的不变测度。此外，借助相应Kolmogorov方程的时间无关正则性估计，数值不变测度与原始不变测度之间的弱误差可达到一阶精度。数值实验验证了我们的理论结果。