Langevin equations and stochastic integrals

Abstract

The ambiguity of stochastic integrals involved in Langevin equations is removed by the postulate of invariance with respect to nonlinear transformations of the coordinates. The Stratonovich sense of the integrals, which is imposed thereby, is also strongly suggested by stability considerations requiring small changes of the solutions whenever the perturbations are changed by a small amount. The associated Fokker-Planck equation must include the “spurious drift” which arises from the transition from the Stratonovich to the Itô sense of the Langevin equations and describes one aspect of the systematic motion due to nonconstant fluctuations.