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 (q) double over dot +e(2)(q)(q)over dot = -del 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 Stratonovich or Ito.