Brownian Motion on the Star Like Quantum Graphs
S.A. Molchanov, M. Paul
2025, v.31, Issue 1, 3-20
ABSTRACT
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 Kirchhoff's) gluing conditions. The limit theorems for the diffusion on such a graph, especially if N → ∞, are significantly different from the classical case N = 2 (full axis). These theorems, among others, contain the generalization of Lévy Arcsine law for Brownian motion and Erdős-Rényi theorem on Maxwell-Boltzmann statistics.
Keywords: Brownian motion, Transition probability, Quantum graphs, Star Laplacian, Spectrum, Absolute continuous spectrum
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