Arnold P
Department of Physics, University of Virginia, Charlottesville, Virginia 22901, USA.
Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics. 2000 Jun;61(6 Pt A):6091-8. doi: 10.1103/physreve.61.6091.
A Langevin equation with multiplicative noise is an equation schematically of the form dq/dt=-F(q)+e(q)xi, where e(q)xi is Gaussian white noise whose amplitude e(q) depends on q itself. Such equations are ambiguous, and depend on the details of one's convention for discretizing time when solving them. I show that these ambiguities are uniquely resolved if the system has a known equilibrium distribution exp[-V(q)/T] and if, at some more fundamental level, the physics of the system is reversible. I also discuss a simple example where this happens, which is the small frequency limit of Newton's equation &quml;+e(2)(q)&qdot;=-nablaV(q)+e(-1)(q)xi with noise and a q-dependent damping term. The resolution does not correspond to simply interpreting naive continuum equations in a standard convention, such as Stratonovich or Itoinsertion mark.
一个带有乘性噪声的朗之万方程是一个形式上大致为dq/dt = -F(q) + e(q)ξ的方程，其中e(q)ξ是高斯白噪声，其幅度e(q)取决于q本身。这类方程具有模糊性，并且在求解时依赖于离散时间的具体约定细节。我表明，如果系统具有已知的平衡分布exp[-V(q)/T]，并且在某些更基本的层面上系统的物理过程是可逆的，那么这些模糊性就能得到唯一解决。我还讨论了一个发生这种情况的简单例子，即牛顿方程&quml; + e(2)(q)&qdot; = -∇V(q) + e(-1)(q)ξ带有噪声和一个依赖于q的阻尼项的小频率极限情况。这种解决方式并不等同于简单地按照标准约定（如斯特拉托诺维奇或伊藤约定）来解释朴素的连续统方程。
