In this paper, we propose a new class of splitting methods to solve the stochastic Langevin equation, which can simultaneously preserve the ergodicity and exponential integrability of the original equation. The central idea is to extract a stochastic subsystem that possesses the strict dissipation from the original equation, which is inspired by the inheritance of the Lyapunov structure for obtaining the ergodicity. We prove that the exponential moment of the numerical solution is bounded, thus validating the exponential integrability of the proposed methods. Further, we show that under moderate verifiable conditions, the methods have the first-order convergence in both strong and weak senses, and we present several concrete splitting schemes based on the methods. The splitting strategy of methods can be readily extended to construct conformal symplectic methods and high-order methods that preserve both the ergodicity and the exponential integrability, as demonstrated in numerical experiments. Our numerical experiments also show that the proposed methods have good performance in the long-time simulation.

翻译：本文提出了一类用于求解随机朗之万方程的新型分裂方法，该方法能同时保持原方程的遍历性与指数可积性。其核心思想是从原方程中提取出一个具有严格耗散性的随机子系统，这一思路源于为获得遍历性而对李雅普诺夫结构的继承。我们证明了数值解的指数矩是有界的，从而验证了所提方法的指数可积性。进一步，我们证明了在中等可验证条件下，该方法在强收敛与弱收敛意义上均具有一阶收敛性，并给出了基于该方法的几种具体分裂格式。如数值实验所示，该方法的分裂策略可轻松扩展至构建共形辛方法以及同时保持遍历性与指数可积性的高阶方法。我们的数值实验也表明，所提方法在长时间模拟中具有良好的性能。