From snapping out Brownian motions to Walsh's spider processes on star-like graphs

TL;DR

This paper demonstrates that snapping-out Brownian motions with high permeability effectively approximate Walsh's spider processes on star-like graphs, linking Brownian motion perturbations to complex network processes.

Contribution

It establishes a rigorous connection between snapping-out Brownian motions and Walsh's spider processes on star graphs, providing a new approximation approach.

Findings

Snapping-out Brownian motions approximate Walsh's spider processes as permeability increases.

The process can be viewed as Brownian motion with a semi-permeable membrane at the graph center.

Theoretical proof based on matrix analysis supports the approximation.

Abstract

By analyzing matrices involved, we prove that a snapping-out Brownian motion with large permeability coefficients is a good approximation of Walsh's spider process on the star-like graph $K_{1,k}$. Thus, the latter process can be seen as a Brownian motion perturbed by a trace of semi-permeable membrane at the graph's center.