turkmath.org

Poincare's invariance principle for Hamiltonian flows implies Kelvin's principle for solution to Incompressible Euler Equation. Iyer-Constantin Circulation Theorem offers a stochastic analog of Kelvin's principle for Navier-Stokes Equation. Weakly symplectic diusions are dened to produce stochastically symplectic flows in a systematic way. With the aid of symplectic diffusions, one can produce a family of martingales associated with solutions to Navier-Stokes Equation that in turn can be used to prove Iyer-Constantin Circulation Theorem. I also discuss how probabilistic ideas can be used to give new proofs for some classical results about solutions of Navier-Stokes Equation.