Stochastic Equations of Motion for an Incompressible Viscous Fluid

Experimental results seem to suggest that the nature of some hydrodynamical phenomena calls for their statistical or stochastic formulation. The main focus of our discussion will be on a 2-dimensional stochastic vorticity equation for an incompressible homogeneous fluid. We consider a signed measure valued stochastic partial differential equation for a vorticity process based on the Skorohod-Ito evolution of a system of N randomly moving point vortices. This approach provides an interesting alternative to some known Navier-Stokes models perturbed by external random forces. We then pose a nonlinear filtering problem associated with the stochastic evolution of vorticity and derive a corresponding Fujisaki-Kallianpur-Kunita stochastic differential equation for the filter.