Optimal Trapping for Brownian Motion: a Nonlinear Analogue of the Torsion Function

Abstract

We study the problem of maximizing the expected lifetime of drift diffusion in a bounded domain. More formally, we consider the PDEin Ω subject to Dirichlet boundary conditions for b L ∞ fixed. We show that, in any given C 2 −domain Ω, the vector field maximizing the expected lifetime is (nonlinearly) coupled to the solution and satisfies b = − b L ∞ ∇u/|∇u| which reduces the problem to the study of the nonlinear PDEWe believe that this PDE is a natural and interesting nonlinear analogue of the torsion function. We prove that, for fixed volume, ∇u L 1 and ∆u L 1 are maximized if Ω is the ball (the ball is also known to maximize u L p for p ≥ 1 from a result of Hamel & Russ).2010 Mathematics Subject Classification. 35B51, 49K20 (primary) and 60J60 (secondary).

Bounds for exit times of Brownian motion and the first Dirichlet eigenvalue for the Laplacian

“…For applications to the study of heat flow and conduction, exit time moments, torsional rigidity, the study of minimal sub-manifolds, and optimal trapping of Brownian motion and gradient estimates, we refer the reader to [28,27,40,43,46,47,53,26]. In [58], the author obtains a spectral bound for the torsion function of symmetric stable processes that is asymptotically sharp for large dimension.…”