  We study the multichannel scattering in the classical three-body system and
show that the problem can be formulated as a motion of the point mass on a
curved hyper-surface of the energy of the body-system. It is proved that the
local coordinate system on curved space produces additional symmetries which
along with known integrals of motion allow to reduce the initial problem to the
system of the sixth order. Assuming, that the metric of the curved space has a
random component, we derive the system of \emph{stochastic differential
equations} (SED) describing the classical motion of the three-body system
taking into account the influence of random forces of various origin and in
particular the quantum fluctuations. Using SDEs of motion, we obtain the
partial differential equation of the second order describing the probability
distribution of the point mass in the momentum representation. It is shown,
that the equation for the probability distribution is solved jointly with the
classical equations, which in turn are responsible for the topological
peculiarities of tubes of quantum currents and transitions between asymptotic
channels. The latter allows to solve the problem of the limiting transition
from the quantum region to the region of classical chaotic motion (Poincar\'e
region) without violating the analogue of Arnold's theorem on quantum mapping.
The expression characterizing a measure of deviation of the quantum
probabilistic currents and thus the appearance of quantum chaos in a dynamical
system is determined.
