  We present a perturbation theory by extending a prescription due to Feynman
for computing the probability density function for the random flight motion.
The method can be applied to a wide variety of otherwise difficult
circumstances. The series for the exact moments, if not the distribution
itself, for many important cases can be summed for arbitrary times. As
expected, the behavior at early time regime, for the sample processes
considered, deviate significantly from diffusion theory; a fact with important
consequences in various applications such as financial physics. A half dozen
sample problems are solved starting with one posed by Feynman who originally
solved it only to first order. Another illustrative case is found to be a
physically more plausible substitute for both the Uhlenbeck-Ornstein and the
Wiener processes. The remaining cases we've solved are useful for applications
in regime-switching and isotropic scattering of light in turbid media. It is
demonstrated that under isotropic random flight with invariant-speed, the
description of motion in higher dimensions is recursively related to either the
one or two dimensional movement. Also for this motion, we show that the
solution heretofore presumed correct in one dimension, applies only to the case
we've named the "Shooting Gallery" problem; a special case of the full problem.
A new class of functions dubbed "Damped-exponential-integrals" are also
identified.
