Exit times of diffusions with incompressible drift

Abstract

Let $Ω\subset\mathbb R^n$ be a bounded domain and for $x\inΩ$ let $τ(x)$ be the expected exit time from $Ω$ of a diffusing particle starting at $x$ and advected by an incompressible flow $u$. We are interested in the question which flows maximize $\|τ\|_{L^\infty(Ω)}$, that is, they are most efficient in the creation of hotspots inside $Ω$. Surprisingly, among all simply connected domains in two dimensions, the discs are the only ones for which the zero flow $u\equiv 0$ maximises $\|τ\|_{L^\infty(Ω)}$. We also show that in any dimension, among all domains with a fixed volume and all incompressible flows on them, $\|τ\|_{L^\infty(Ω)}$ is maximized by the zero flow on the ball.