Invariant measure selection by noise: An Example
  We consider a deterministic system with two conserved quantities and infinity many invariant measures. However the systems possess a unique invariant measure when enough stochastic forcing and balancing dissipation are added. We then show that as the forcing and dissipation are removed a unique limit of the deterministic system is selected. The exact structure of the limiting measure depends on the specifics of the stochastic forcing. 
Jonathan C. Mattingly and Etienne Pardoux
math.PR math.DS physics.flu-dyn