caa a22 4500 467911088 CHVBK 20180406152849.0 cr unu---uuuuu 170328e20061201xx s 000 0 eng 10.1007/s11232-006-0153-y doi (NATIONALLICENCE)springer-10.1007/s11232-006-0153-y Fainberg V. Lebedev Physics Institute, Moscow, Russia aut Connection between the Fokker-Planck-Kolmogorov and nonlinear Langevin equations [Elektronische Daten] [V. Fainberg] 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. Springer Science+Business Media, Inc., 2006 Langevin equation nationallicence Euclidean space nationallicence Theoretical and Mathematical Physics Kluwer Academic Publishers-Consultants Bureau 149/3(2006-12-01), 1710-1725 0040-5779 149:3<1710 2006 149 11232 https://doi.org/10.1007/s11232-006-0153-y text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s11232-006-0153-y text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Fainberg V. Lebedev Physics Institute, Moscow, Russia aut NATIONALLICENCE 773 0- Theoretical and Mathematical Physics Kluwer Academic Publishers-Consultants Bureau 149/3(2006-12-01), 1710-1725 0040-5779 149:3<1710 2006 149 11232 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer