A probabilistic representation for the vorticity of a three-dimensional viscous fluid and for general systems of parabolic equations

A probabilistic representation formula for general systems of linear parabolic equations, coupled only through the zero-order term, is given. On this basis, an implicit probabilistic representation for the vorticity in a three-dimensional viscous fluid (described by the Navier–Stokes equations) is carefully analysed, and a theorem of local existence and uniqueness is proved. The aim of the probabilistic representation is to provide an extension of the Lagrangian formalism from the non-viscous (Euler equations) to the viscous case. As an application, a continuation principle, similar to the Beale–Kato–Majda blow-up criterion, is proved.