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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
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
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
Linking options:
https://www.mathnet.ru/eng/tmf5539https://doi.org/10.4213/tmf5539 https://www.mathnet.ru/eng/tmf/v149/i3/p483
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Abstract page: | 1491 | Full-text PDF : | 723 | References: | 60 | First page: | 2 |
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