A kinetic model approximation of Walsh's spider process on the infinite star-like graph

TL;DR

This paper develops a kinetic model approximation for Walsh's spider process on an infinite star-like graph, combining deterministic motions with stochastic perturbations, and proves convergence to the Walsh's spider process under diffusing scaling.

Contribution

It introduces a kinetic model approximation for Walsh's spider process on star-like graphs and establishes its convergence in the diffusing limit.

Findings

Proves convergence of the kinetic model to Walsh's spider process.

Models stochastic perturbations at the graph's center and between edges.

Provides a rigorous mathematical framework for the approximation.

Abstract

We consider processes of deterministic motions on $k$ copies of the star-like graph $S_k= K_{1,k}$ with $k$ edges which are perturbed by two stochastic mechanisms: one caused by interfaces located at the graphs' centers, the other describing jumps between different copies of the same edge. We prove that diffusing scaling of these processes leads in the limit to the Walsh's spider process on $S_k$.