A Stochastic Representation Of The Two-dimensional Quasi-geostrophic Equation

In this paper we derive a probabilistic representation of the two-dimensional quasi-geostrophic equation based on stochastic Lagrangian paths.The particle trajectories obey SDEs driven by a uniformal Wiener process.This method can be extended to formulate stochastic representations of related hydrodynamic-type and quasi-linear equations,including Lagrangian-averaged Navier-Stokes alpha models and surface quasi-geostrophic equation(SQG),and the case of dissipative quasi-geostrophic equation is also considered,and provide a simple proof.In this paper,we also give the local existence theorems and their proofs of the solution of stochastic representation,finally,we consider the global existence of the solution of stochastic representation.