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Publication 09-CNA-16

Exit Times of Diffusions with Incompressible Drift

Gautam Iyer
Carnegie Mellon University
Department of Mathematical Sciences
Pittsburgh, PA 15213
gi1242@andrew.cmu.edu

lexei Novikov
Pennsylvania State University
Department of Mathematics
State College, PA 16802
anovikov@math.psu.edu

Lenya Ryzhik
Stanford University
Department of Mathematics
Chicago, IL 60637
ryzhik@math.uchicago.edu

Andrej Zlatos
University of Chicago
Department of Mathematics
Chicago, IL 60637
andrej@math.uchicago.edu

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.

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