Passive scalar equation in a turbulent incompressible Gaussian velocity field

The time evolution of a passive scalar in a turbulent homogeneous incompressible Gaussian flow is considered. The turbulent nature of the flow results in non-smooth coefficients of the corresponding evolution equation. A strong solution (in the probabilistic sense) of the equation is constructed by using the Wiener Chaos expansion, and properties of the solution are studied. In particular, a certain $L_p$-regularity of the solution and a representation formula of Feynman–Kac type (or a Lagrangian formula) are among the results obtained. The results can be applied to both viscous and conservative flows.