Stochastic continuity equation and related processes

We consider the processes defined by a Langevin equation and the associated continuity equation. The average of the density function, solution of the continuity equation, satisfies the Fokker-Planck equation. For a volume preserving vector field the same equation is satisfied by the average of the integer powers of the density, which are the moments of the related probability density. For a generic vector field the Fokker-Planck equation for the moments is slightly modified. We first illustrate the problem in the simple case of a free particle subject to a white noise, since the averages can be computed by an elementary procedure using the factorization property of the correlation functions of the noise. The probabilistic meaning of the moments is discussed and the comparison between the analytical results and the numerical simulation is shown. The case of a generic Langevin equation is treated by computing the averages via a Dyson expansion after observing that, for a volume preserving vector field, any power of the density function satisfies the same continuity equation with the appropriate initial conditions. As a consequence the results obtained for the free particle are easily extended. An alternative approach is based on the characteristics of the continuity equation; the probability density of this process in an extended phase space still satisfies a Fokker-Planck equation and its moments coincide with the previous definition.