随机微分方程在离散化下的Smoluchowski-Kramers 逼近

Smoluchowski-Kramers approximation for stochastic differential equations under discretization

  • 摘要: 本文研究了由粒子在力场中的运动所描述的动力系统在时间离散化下的Smoluchowski-Kramers逼近。我们证明了当使用漂移项隐-Euler-Maruyama格式进行离散化时,Smoluchowski-Kramers逼近成立,并得到了收敛速度。特别地,离散化系统的解在均方意义下收敛于一阶方程的解,且它不依赖于质量\mu和步长h趋于0的阶数。

     

    Abstract: The Smoluchowski-Kramers approximation of discrete time dynamical system is considered, where the system is described by the motion of a particle in a force field. It is shown that the Smoluchowski-Kramers approximation holds when a drift-implicit Euler-Maruyama scheme is used for the discretization, moreover the convergence rate is obtained. In particular, the solution of discretized system converges to the solution of the first order equation in mean square sense, which does not depend on the order of mass \mu and step size h tend to zero.

     

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