Universal formula for the mean first passage time in planar domains

TL;DR

This paper presents a universal, exact formula for the mean first passage time in planar domains, valid for arbitrary space-dependent diffusion, revealing the harmonic measure as the key parameter and challenging traditional scaling assumptions.

Contribution

The authors derive a general exact formula for MFPT in planar domains using conformal mapping, highlighting the harmonic measure's role and providing a new perspective on escape problems.

Findings

MFPT formula is valid for arbitrary diffusion coefficients.

Harmonic measure, not perimeter, is the natural small parameter.

Scaling of MFPT is altered near the escape region.

Abstract

We derive a general exact formula for the mean first passage time (MFPT) from a fixed point inside a planar domain to an escape region on its boundary. The underlying mixed Dirichlet-Neumann boundary value problem is conformally mapped onto the unit disk, solved exactly, and mapped back. The resulting formula for the MFPT is valid for an arbitrary space-dependent diffusion coefficient, while the leading logarithmic term is explicit, simple, and remarkably universal. In contrast to earlier works, we show that the natural small parameter of the problem is the harmonic measure of the escape region, not its perimeter. The conventional scaling of the MFPT with the area of the domain is altered when diffusing particles are released near the escape region. These findings change the current view of escape problems and related chemical or biochemical kinetics in complex, multiscale, porous or fractal domains, while the fundamental relation to the harmonic measure opens new ways of computing and interpreting MFPTs.