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 $\Vert \tau {\Vert }_{L^{\infty }(\Omega )}$, 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 $u\equiv 0$ maximises $\Vert \tau {\Vert }_{L^{\infty }(\Omega )}$. We also show that in any dimension, among all domains with a fixed volume and all incompressible flows on them, $\Vert \tau {\Vert }_{L^{\infty }(\Omega )}$ is maximized by the zero flow on the ball.

15 pages, 1 figure