Brownian Motion on The Spider Like Quantum Graphs

TL;DR

This paper analyzes Brownian motion on spider-like quantum graphs, revealing unique limit behaviors as the number of branches increases and exploring spectral properties of associated Laplacians.

Contribution

It provides new probabilistic limit theorems for diffusion on spider graphs and investigates spectral measures and Fourier transforms of the spider Laplacian.

Findings

Limit theorems for diffusion as N approaches infinity

Spectral measure properties of the spider Laplacian

Differences from classical cases with N=2

Abstract

The paper contains the probabilistic analysis of the Brownian motion on the simplest quantum graph, spider: a system of N-half axis connected only at the graph's origin by the simplest (so-called Kirchhoff's) gluing conditions. The limit theorems for the diffusion on such a graph, especially if $N \to \infty$ are significantly different from the classical case $N = 2$ (full axis). Additional results concern the properties of the spectral measure of the spider Laplacian and the corresponding generalized Fourier transforms. The continuation of the paper will contain the study of the spectrum for the class of Schr\"odinger operators on the spider graphs: Laplacian perturbed by unbounded potential and related phase transitions.