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Rules of calculus in the path integral representation of white noise Langevin equations: the Onsager-Machlup approach
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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 It\=o prescription ($dB^2=dt$) 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.
Forward citations
Cited by 2 Pith papers
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Path integral analysis of Schr\"odinger-type eigenvalue problems in the complex plane: Establishing the relation between instantons and resonant states
Path integrals on complex contours that terminate in prescribed Stokes sectors yield spectral formulas for resonant energies, explaining why the instanton bounce calculation and real-time decay rates agree.
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On the use of the Belopol'skaya-Daletskii representation of a diffusion on a Riemann manifold to construct path integrals
A Belopol'skaya-Daletskii (exponential-map) formulation yields the known scalar-curvature term R/6 in finite-dimensional path integrals for diffusions on Riemannian manifolds.
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