Asymptotic analysis on narrow tubes: narrow escape problems and diffusion processes

TL;DR

This paper analyzes diffusion in narrow tubular domains, deriving asymptotic behaviors and convergence to graph-based diffusions, with applications to escape times and boundary behaviors in complex geometries.

Contribution

It introduces a rigorous asymptotic analysis of diffusion in narrow tubes, connecting geometric perturbations to diffusion processes on graphs with specific boundary conditions.

Findings

Weak convergence of the process to a graph-based diffusion.

Asymptotic formulas for expected escape times.

Exponential distribution of escape times in the limit.

Abstract

This paper investigates a diffusion process in a narrow tubular domain with reflecting boundary conditions, where the geometry serves as a singular perturbation of an underlying graph in $\mathbb{R}^2$ or $\mathbb{R}^3$. The construction incorporates distinct scaling regimes in the neighborhoods of the graph's vertices and edges. We show that, in the limit, the projected process converges weakly to a diffusion process on the graph, with gluing conditions at the vertices that depend on the relative scales of the neighborhoods. Our analysis relies on a detailed understanding of the narrow escape problem in domains with bottlenecks. In particular, we rigorously derive the asymptotic behavior of the expected escape time, establish the asymptotic exponential distribution of escape times and obtain exit place estimates, results that may be of independent interest.