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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Journal of Physics A General Physics, Institute of Physics (IOP), 2017, 50 (34), pp.345001. 〈http://iopscience.iop.org/article/10.1088/1751-8121/aa7dd6/meta〉. 〈10.1088/1751-8121/aa7dd6〉
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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, Institute of Physics (IOP), 2017, 50 (34), pp.345001. 〈http://iopscience.iop.org/article/10.1088/1751-8121/aa7dd6/meta〉. 〈10.1088/1751-8121/aa7dd6〉. 〈hal-01508497〉

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