Stochastically symplectic maps and their applications to the Navier–Stokes equation

Poincaré's invariance principle for Hamiltonian flows implies Kelvin's principle for solution to Incompressible Euler equation. Constantin–Iyer Circulation Theorem offers a stochastic analog of Kelvin's principle for Navier–Stokes equation. Weakly symplectic diffusions are defined to produce stochastically symplectic flows in a systematic way. With the aid of symplectic diffusions, we produce a family of martigales associated with solutions to Navier–Stokes equation that in turn can be used to prove Constantin–Iyer Circulation Theorem. We also review some basic facts in symplectic and contact geometry and their applications to Euler equation.