Moments of the distributions in probabilistic dynamics

Let π(i,Image ,t) be the probability density for a physical system to be in a component state i with physical variables Image at time t. Its evolution is given by the Chapman-Kolmogorov equation, which is only analytically solvable in very simple cases. In this paper, we show how to obtain the first moments in order of the distributions. These moments are solutions of a large and coupled differential system that we have to close first. A specific algorithm is presented for this problem and is illustrated on different applications.