The paper is focused on analyzing the conservation issues of stochastic 𝜃-methods when applied to nonlinear damped stochastic oscillators. In particular, we are interested in reproducing the long-term properties of the continuous problem over its discretization through stochastic 𝜃-methods, by preserving the correlation matrix. This evidence is equivalent to accurately maintaining the stationary density of the position and the velocity of a particle driven by a nonlinear deterministic forcing term and an additive noise as a stochastic forcing term. The provided analysis relies on a linearization of the nonlinear problem, whose effectiveness is proved theoretically and numerically confirmed.

本文重点分析了随机𝜃方法在非线性阻尼随机振荡器中的守恒问题。尤其是，我们有兴趣通过保留相关矩阵，通过随机𝜃方法在离散化上重现连续问题的长期性质。该证据等效于精确地维持由非线性确定性强迫项和加性噪声作为随机强迫项驱动的粒子的位置的固定密度和速度。提供的分析依赖于非线性问题的线性化，其有效性在理论上和数值上得到了证明。