Diffusion processes in thin tubes and their limits on graphs

TL;DR

This paper studies diffusion processes in shrinking tubular domains and their convergence to processes on graphs, using probabilistic methods to characterize the limit with Kirchhoff boundary conditions.

Contribution

It introduces a probabilistic approach to analyze the limit of diffusion processes in thin tubes shrinking to graphs, establishing uniqueness and characterization of the limit process.

Findings

Existence of a unique limit process as tubes shrink to graphs.

Characterization of the limit process by a differential generator with Kirchhoff conditions.

Application of probabilistic methods to boundary and shrinking problems.

Abstract

The present paper is concerned with diffusion processes running on tubular domains with conditions on nonreaching the boundary, respectively, reflecting at the boundary, and corresponding processes in the limit where the thin tubular domains are shrinking to graphs. The methods we use are probabilistic ones. For shrinking, we use big potentials, respectively, reflection on the boundary of tubes. We show that there exists a unique limit process, and we characterize the limit process by a second-order differential generator acting on functions defined on the limit graph, with Kirchhoff boundary conditions at the vertices.