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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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arxiv 1704.03501 v2 pith:3ASLLLHS submitted 2017-04-11 cond-mat.stat-mech math-phmath.MPmath.PR

classification cond-mat.stat-mechmath-phmath.MPmath.PR
keywords langevinrepresentationcalculuschangesequationsnoiseonsager-machluppath-integral
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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.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Path integral analysis of Schr\"odinger-type eigenvalue problems in the complex plane: Establishing the relation between instantons and resonant states

    hep-th 2025-07 conditional novelty 6.0 of 10

    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.

  2. On the use of the Belopol'skaya-Daletskii representation of a diffusion on a Riemann manifold to construct path integrals

    cond-mat.stat-mech 2026-07 conditional novelty 4.0 of 10

    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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