  We present certain mathematical aspects of an information method which was
formulated in an attempt to investigate diffusion phenomena. We imagine a
regular dynamical hamiltonian systems under the random perturbation of thermal
(molecular) noise and chaotic motion. The random effect is taken into account
via the uncertainty of irregular dynamic process produced in this way. This
uncertainty due to different paths between two phase points is measured by a
path information which is maximized in connection with the action defined
originally for the unperturbed regular hamiltonian systems. The obtained
transition probability depends exponentially on this action. The usefulness of
this information method has been demonstrated by the derivation of diffusion
laws without the usual assumptions. In this work, some essential mathematical
aspects of this irregular dynamics is reviewed. It is emphasized that the
classical action principle for single least action path is no more valid and
the formalism of classical mechanics for regular hamiltonian systems is no more
exact for irregular hamiltonian dynamics. There is violation of the fundamental
laws of mechanics by randomly perturbed hamiltonian systems. However, the
action principle is always present for the ensemble of paths through the
average action. This average action principle leads to a formalism of
stochastic mechanics in which, in spite of the violation of fundamental laws,
the mathematical form of classical mechanics can be recovered by a
consideration of the statistical averaging of the dynamics.
