Exit Times of Diffusions with Incompressible Drift

Abstract

Let $\Omega\subset\mathbb{R}^n$ be a bounded domain, and for $x\in\Omega$ let $\tau(x)$ be the expected exit time from $\Omega$ of a diffusing particle starting at x and advected by an incompressible flow u. We are interested in the question which flows maximize $\|\tau\|_{L^\infty(\Omega)}$, that is, they are most efficient in the creation of hotspots inside $\Omega$. Surprisingly, among all simply connected domains in two dimensions, the discs are the only ones for which the zero flow $u\equiv0$ maximizes $\|\tau\|_{L^\infty(\Omega)}$. We also show that in any dimension, among all domains with a fixed volume and all incompressible flows on them, $\|\tau\|_{L^\infty(\Omega)}$ is maximized by the zero flow on the ball.