  We investigate exit times from domains of attraction for the motion of a
self-stabilized particle traveling in a geometric (potential type) landscape
and perturbed by Brownian noise of small amplitude. Self-stabilization is the
effect of including an ensemble-average attraction in addition to the usual
state-dependent drift, where the particle is supposed to be suspended in a
large population of identical ones. A Kramers' type law for the particle's exit
from the potential's domains of attraction and a large deviations principle for
the self-stabilizing diffusion are proved. It turns out that the exit law for
the self-stabilizing diffusion coincides with the exit law of a potential
diffusion without self-stabilization and a drift component perturbed by average
attraction. We show that self-stabilization may substantially delay the exit
from domains of attraction, and that the exit location may be completely
different.
