On the stochastic least action principle for the navier-stokes equation

Abstract

In this paper we extend the class of stochastic processes allowed to represent solutions of the Navier-Stokes equation on the two dimensional torus to certain non-Markovian processes which we call admissible. More precisely, using the variations of Ref. [3], we provide a criterion for the associated mean velocity field to solve this equation. Due to the fluctuations of the shift a new term of pressure appears which is of purely stochastic origin. We provide an alternative formulation of this least action principle by means of transformations of measure. Within this approach the action is a function of the law of the processes, while the variations are induced by some translations on the space of the divergence free vector fields. Due to the renormalization in the definition of the cylindrical Brownian motion, our action is only related to the relative entropy by an inequality. However we show that, if we cut the high frequency modes, this new approach provides a least action principle for the Navier-Stokes equation based on the relative entropy.