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 ∂Ω, a non-linear branching process ($Y_t; t ≥ 0$). More precisely: E_{ω0} (h,Y_t) = (ω, h), for any test function $h$, where ω = 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 ∂Ω; this stochastic phenomenon corresponds to the creation-annihilation of vortex localized at the boundary of Ω.