Branching process associated with 2d-Navier Stokes equation

Abstract

$\Omega$ being a bounded open set in ${\mathbb{R}}^2$, with regular boundary, we associate with Navier–Stokes equation in $\Omega$ where the velocity is null on $\partial \Omega$, a non-linear branching process ($Y_t;t\ge 0$). More precisely: $E_{\omega 0}(h,Y_t)=(\omega ,h)$, for any test function $h$, where $\omega =\mathrm{rot}u$, $u$ denotes the velocity solution of Navier–Stokes equation. The support of the random measure $Y_t$ increases or decreases of one unit when the underlying process hits $\partial \Omega$; this stochastic phenomenon corresponds to the creation-annihilation of vortex localized at the boundary of $\Omega$.