  Markovian diffusion processes yield a system of conservation laws which
couple various conditional expectation values (local moments). Solutions of
that closed system of deterministic partial differential equations stand for a
regular alternative to erratic (irregular) sample paths that are associated
with weak solutions of the primordial stochastic differential equations. We
investigate an issue of local characteristics of motion in the non-Gaussian
context, when moments of the probability measure may not exist. A particular
emphasis is put on jump-type stochastic processes with the
Ornstein-Uhlenbeck-Cauchy process as a fully computable exemplary case.
