Local characteristics of random motion

Markovian diffusion processes yield a system of conservation laws which cou ple various conditional expectation values (local moments). Solutions of th at closed system of deterministic partial differential equations stand for a regular alternative to erratic (irregular) sample paths that are associat ed with weak solutions of the original stochastic differential equations. W e investigate an issue of local characteristics of motion in the non-Gaussi an context, when moments of the probability measure may not exist. A partic ular emphasis is put on jump-type stochastic processes with the Omstein-Uhl enbeck-Cauchy process as a fully computable exemplary case.