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Teoreticheskaya i Matematicheskaya Fizika, 2006, Volume 149, Number 3, Pages 483–501
DOI: https://doi.org/10.4213/tmf5539
(Mi tmf5539)
 

This article is cited in 3 scientific papers (total in 3 papers)

Connection between the Fokker–Planck–Kolmogorov and nonlinear Langevin equations

V. Ya. Fainberg

P. N. Lebedev Physical Institute, Russian Academy of Sciences
Full-text PDF (535 kB) Citations (3)
References:
Abstract: We recall the general proof of the statement that the behavior of every holonomic nonrelativistic system can be described in terms of the Langevin equation in Euclidean $($imaginary$)$ time such that for certain initial conditions, the different stochastic correlators $($after averaging over the stochastic force$)$ coincide with the quantum mechanical correlators. The Fokker–Planck–Kolmogorov $($FPK$)$ equation that follows from this Langevin equation is equivalent to the Schrödinger equation in Euclidean time if the Hamiltonian is Hermitian, the dynamics are described by potential forces, the vacuum state is normalizable, and there is an energy gap between the vacuum state and the first excited state. These conditions are necessary for proving the limit and ergodic theorems. For three solvable models with nonlinear Langevin equations, we prove that the corresponding Schrödinger equations satisfy all the above conditions and lead to local linear FPK equations with the derivative order not exceeding two. We also briefly discuss several subtle mathematical questions of stochastic calculus.
Keywords: Langevin equation, Euclidean space.
Received: 20.07.2006
English version:
Theoretical and Mathematical Physics, 2006, Volume 149, Issue 3, Pages 1710–1725
DOI: https://doi.org/10.1007/s11232-006-0153-y
Bibliographic databases:
Language: Russian
Citation: V. Ya. Fainberg, “Connection between the Fokker–Planck–Kolmogorov and nonlinear Langevin equations”, TMF, 149:3 (2006), 483–501; Theoret. and Math. Phys., 149:3 (2006), 1710–1725
Citation in format AMSBIB
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  • https://www.mathnet.ru/eng/tmf5539
  • https://doi.org/10.4213/tmf5539
  • https://www.mathnet.ru/eng/tmf/v149/i3/p483
  • This publication is cited in the following 3 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Теоретическая и математическая физика Theoretical and Mathematical Physics
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    References:60
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