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)ξ, where e(q)ξ 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 $q{}{+e}²(q)q ̇ \.{}{}={-}{{∇}}V(q{)+e}^{{-}1}(q){{ξ}}$ 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 It\^o.