Langevin equations with multiplicative noise: resolution of time discretization ambiguities for equilibrium systems

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 d^2q/dt^2

• e^2(q) dq/dt = - grad V(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 Stratanovich or Ito. [One application of Langevin equations with multiplicative noise is to certain effective theories for hot, non-Abelian plasmas.]