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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 particlestarting at $x$ and advected by an incompressible flow $u$. We are interestedin 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 discsare the only ones for which the zero flow $u\equiv 0$ maximises$\|\tau\|_{L^\infty(\Omega)}$. We also show that in any dimension, among alldomains with a fixed volume and all incompressible flows on them,$\|\tau\|_{L^\infty(\Omega)}$ is maximized by the zero flow on the ball.

Exit times of diffusions with incompressible drift