Periodic phenomena such as oscillation have been studied for many years. In this paper, we verify the stochastic version of Levinson’s conjecture, which confirmed the existence of stochastic periodic solutions for second order Newtonian systems with dissipativeness. First, we provide a stochastic Duffing’s equation to display our result. Then, we apply Wong–Zakai approximation method and Lyapunov’s method to stochastic second order Newtonian systems driven by Brownian motions. With the help of Horn’s fixed point theorem, we prove that this kind of systems is stochastic dissipative and admits periodic solutions in distribution.

中文翻译：

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随机牛顿系统分布中周期解的存在性

诸如振荡之类的周期性现象已被研究多年。在本文中，我们验证了 Levinson 猜想的随机版本，该猜想证实了具有耗散性的二阶牛顿系统的随机周期解的存在。首先，我们提供了一个随机 Duffing 方程来显示我们的结果。然后，我们将Wong-Zakai 近似方法和Lyapunov 方法应用于由布朗运动驱动的随机二阶牛顿系统。借助霍恩不动点定理，我们证明了这种系统是随机耗散的，并且在分布上允许周期解。