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The definition and manipulation of Langevin equations with multiplicativewhite noise require special care (one has to specify the time discretisationand a stochastic chain rule has to be used to perform changes of variables).While discretisation-scheme transformations and non-linear changes of variablecan be safely performed on the Langevin equation, these same transformationslead to inconsistencies in its path-integral representation. We identify theirorigin and we show how to extend the well-known It\=o prescription ($dB^2=dt$)in a way that defines a modified stochastic calculus to be used inside thepath-integral representation of the process, in its Onsager-Machlup form.

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