Random vortex dynamics via functional stochastic differential equations

TL;DR

This paper introduces a novel 3D random vortex dynamics system linked to Navier-Stokes equations, enabling new numerical methods for simulating incompressible fluid flows via Monte Carlo techniques.

Contribution

It develops a new 3D stochastic vortex model coupled with a functional differential equation, expanding the mathematical toolkit for nonlinear fluid dynamics problems.

Findings

Provides a closed-form 3D vortex system equivalent to Navier-Stokes

Develops duality of conditional distributions of Taylor diffusions

Introduces a Feynman-Kac formula for nonlinear parabolic equations

Abstract

In this paper we present a novel, closed three-dimensional (3D) random vortex dynamics system, which is equivalent to the Navier--Stokes equations for incompressible viscous fluid flows. The new random vortex dynamics system consists of a stochastic differential equation which is, in contrast with the two-dimensional random vortex dynamics equations, coupled with a finite-dimensional ordinary functional differential equation. This new random vortex system paves the way for devising new numerical schemes (random vortex methods) for solving three-dimensional incompressible fluid flow equations by Monte Carlo simulations. In order to derive the 3D random vortex dynamics equations, we have developed two powerful tools: the first is the duality of the conditional distributions of a couple of Taylor diffusions, which provides a path space version of integration by parts; the second is a forward type Feynman--Kac formula representing solutions to nonlinear parabolic equations in terms of functional integration. These technical tools and the underlying ideas are likely to be useful in treating other nonlinear problems.