Rules of calculus in the path integral representation of white noise Langevin equations: the Onsager–Machlup approach

The definition and manipulation of Langevin equations with multiplicative white noise require special care (one has to specify the time discretisation and a stochastic chain rule has to be used to perform changes of variables). While discretisation-scheme transformations and non-linear changes of variable can be safely performed on the Langevin equation, these same transformations lead to inconsistencies in its path-integral representation. We identify their origin and we show how to extend the well-known Ito prescription (dB2=dt) in a way that defines a modified stochastic calculus to be used inside the path-integral representation of the process, in its Onsager–Machlup form.