Local Characteristics of Random Motion

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.