Exit Times of Diffusions with Incompressible Drift

Let Ω ⊂ Rn be a bounded domain, and for x ∈ Ω 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∞(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 ≡ 0 maximizes ‖τ‖L∞(Ω). We also show that in any dimension, among all domains with a fixed volume and all incompressible flows on them, ‖τ‖L∞(Ω) is maximized by the zero flow on the ball.