Connection between the~Fokker--Planck--Kolmogorov and nonlinear Langevin equations

V. Ya. Fainberg

Teoretičeskaâ i matematičeskaâ fizika, Tome 149 (2006) no. 3, pp. 483-501

Résumé

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.

Détail

Mots-clés : Langevin equation
Keywords: Euclidean space.