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

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 ($d{B}^2=dt$) 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.

35 pages, 2 figures; minor modifications

References in corpus (5)

• State-dependent diffusion: thermodynamic consistency and its path integral formulation
• Symmetries of generating functionals of Langevin processes with colored multiplicative noise
• Universal Form of Stochastic Evolution for Slow Variables in Equilibrium Systems
• Langevin dynamics for vector variables driven by multiplicative white noise: a functional formalism
• Path Integral approach to nonequilibrium potentials in multiplicative Langevin dynamics

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