Characterizations of polynomial-Gaussian processes that are Markovian

We consider questions of characterizing a stochastic process X= (Xt, t ≥ 0) by the properties of the first two conditional moments. Our first result is a new version of the classical P. Lévy characterization theorem for martingales. Next we deal with a characterization of processes without continuous trajectories. We consider a special form of the initial state. Namely, we suppose that the r.v. X0 has a polynomial-normal distribution (PND), i.e. the density of X0 is the product of a positive polynomial and a normal density.