Brownian Motion on the Star Like Quantum Graphs

The paper contains the probabilistic analysis of the Brownian motion on the simplest quantum graph, star: a system of N-half axis connected only at the graph's origin by the simplest (so-called Kirchhoffs) gluing conditions. The limit theorems for the diffusion on such a graph, especially if N → ∞ are signi ficantly different from the classical case N = 2 (full axis). These theorems, among others, contain the generalization of Levy Arcsine law for Brownian motion and Erdos-Renyi theorem on Maxwell-Boltzmann statistics.