Symmetries and martingales in a stochastic model for the Navier-Stokes equation

TL;DR

This paper develops a stochastic framework for Navier-Stokes solutions using semi-martingales, linking symmetries to martingales via a weak Noether's theorem, and introduces a least action principle based on relative entropy.

Contribution

It introduces a novel stochastic representation of Navier-Stokes solutions and connects symmetries to martingales through a weak Noether's theorem, with a new least action principle.

Findings

Representation of Navier-Stokes solutions by semi-martingales

Symmetries lead to martingales via a weak Noether's theorem

A least action principle based on relative entropy is established

Abstract

A stochastic description of solutions of the Navier-Stokes equation is investigated. These solutions are represented by laws of finite dimensional semi-martingales and characterized by a weak Euler- Lagrange condition. A least action principle, related to the relative entropy, is provided. Within this stochastic framework, by assuming further symmetries, the corresponding invariances are expressed by martingales, stemming from a weak Noether's theorem.