Backward Stochastic Differential Equations Associated with the Vorticity Equations

TL;DR

This paper develops a stochastic representation for 2D vorticity equations with periodic boundaries, establishing existence and uniqueness of solutions via a non-linear Feynman-Kac formula.

Contribution

It introduces a novel stochastic framework linking backward stochastic differential equations to 2D vorticity equations, extending the Feynman-Kac formula to non-linear PDEs.

Findings

Established a non-linear Feynman-Kac formula for 2D vorticity equations.

Proved global existence and uniqueness of solutions for the associated stochastic problem.

Provided a new probabilistic approach to analyze vorticity dynamics.

Abstract

We derive a non-linear version of the Feynman-Kac formula for the solutions of the vorticity equation in dimension 2 with space periodic boundary conditions. We prove the existence (global in time) and uniqueness for a stochastic terminal value problem associated with the vorticity equation in dimension 2.