The Continuity Equation with Random Velocity Field and the Expectation of Its Solutions

The continuity equation with the random velocity field in the d-dimensional space is studied and the existence of the corresponding family of random evolution operators is proved. In case of the random velocities defined by means of the Gaussian random fields the expectations of the evolution operators are derived and represented in terms of the diffusion operator. The obtained result is applied to the environmental transport problem on a bounded domain. The obtained transport models generalize the conventional transport model defined by means of the diffusion advection equation