Exit times of diffusions with incompressible drift

TL;DR

This paper investigates how incompressible flows influence the maximum expected exit time of diffusing particles in bounded domains, revealing that in 2D discs and higher dimensions, the zero flow maximizes this exit time.

Contribution

It establishes that in two dimensions, discs are uniquely optimal for maximizing exit times under zero flow, and in higher dimensions, the zero flow on a ball maximizes the expected exit time among all incompressible flows.

Findings

Discs are the only simply connected 2D domains where zero flow maximizes exit time.

In any dimension, zero flow on a ball maximizes the maximum expected exit time.

Zero flow is optimal for hotspot creation in the studied setting.

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 $u\equiv 0$ maximises $\|\tau\|_{L^\infty(\Omega)}$. 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.