随机反应扩散方程的宏观约化
Macroscopic reduction for stochastic reaction-diffusion equations
摘要：
耗散随机偏微分方程的宏观行为通常可以用一个有限维系统来描述。本文证明了具有三次非线性项的随机反应扩散方程的宏观约化模型可以通过人为地将系统分成两个不同的慢-快时间部分来构造。平均方法和偏差估计表明，宏观简化模型应该是一个随机常微分方程，它包含了由于非线性相互作用从微观时间尺度传递过来的随机效应。一个例子随机热方程的数值模拟证实了这种随机建模理论的预测。这一理论使我们能够更好地模拟复杂随机系统的长时间动态。
英文摘要：
The macroscopic behavior of dissipative stochastic partial differential equations usually can be described by a finite dimensional system. This article proves that a macroscopic reduced model may be constructed for stochastic reaction-diffusion equations with cubic nonlinearity by artificial separating the system into two distinct slow-fast time parts. An averaging method and a deviation estimate show that the macroscopic reduced model should be a stochastic ordinary equation which includes the random effect transmitted from the microscopic timescale due to the nonlinear interaction. Numerical simulations of an example stochastic heat equation confirms the predictions of this stochastic modelling theory. This theory empowers us to better model the long time dynamics of complex stochastic systems.
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