Probabilistic representation for mild solution of the Navier-Stokes equations

TL;DR

This paper establishes a probabilistic representation formula for mild solutions of the Navier-Stokes equations, linking fluid velocity fields to stochastic processes, thereby providing a new perspective on solving these fundamental equations.

Contribution

It extends the probabilistic formulation of Navier-Stokes equations to mild solutions on  spaces, building on prior work by Iyer and Constantin.

Findings

Probabilistic representation formula for mild solutions established

Provides a new stochastic perspective on Navier-Stokes equations

Enhances understanding of fluid dynamics through stochastic methods

Abstract

This paper is based on a formulation of the Navier-Stokes equations developed by Iyer and Constantin \cite{Cont} , where the velocity field of a viscous incompressible fluid is written as the expected value of a stochastic process. Our contribution is to establish this probabilistic representation formula for mild solutions of the Navier-Stokes equations on $\mathbb{R}^{d} $.