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Rules of calculus in the path integral representation of white noise Langevin equations

Abstract : 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 Itō 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.
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Leticia F. Cugliandolo, Vivien Lecomte. Rules of calculus in the path integral representation of white noise Langevin equations. Journal of Physics A General Physics (1968-1972), Institute of Physics (IOP), 2017, 50 (34), pp.345001. ⟨10.1088/1751-8121/aa7dd6⟩. ⟨hal-01508497⟩

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