  In this paper, we consider a diffusion process pertaining to a chain of
distributed control systems with small random perturbation. The distributed
control system is formed by n subsystems that satisfy an appropriate Hormander
condition, i.e., the second subsystem assumes the random perturbation entered
into the first subsystem, the third subsystem assumes the random perturbation
entered into the first subsystem then was transmitted to the second subsystem
and so on, such that the random perturbation propagates through the entire
distributed control system. Note that the random perturbation enters only in
one of the subsystems and, hence, the diffusion process is degenerate, in the
sense that the backward operator associated with it is a degenerate parabolic
equation. Our interest is to estimate the exit probability with which a
diffusion process (corresponding to a particular subsystem) exits from a given
bounded open domain during a certain time interval. The method for such an
estimate basically relies on the interpretation of the exit probability
function as a value function for a family of stochastic control problems that
are associated with the underlying chain of distributed control systems.
