Universal properties of branching random walks in confined geometries

¹ CEA/Saclay, DEN/DANS/DM2S/SERMA/LTSD - Gif-sur-Yvette, France
² Université Paris-Sud, LPTMS, CNRS (UMR 8626) - 91405 Orsay Cedex, France

Received: 24 June 2014
Accepted: 14 July 2014

Abstract

Characterizing the occupation statistics of random walks through confined geometries amounts to assessing the distribution of the travelled length ℓ and the number of collisions n performed by the stochastic process in a given region, for which remarkably simple Cauchy-like formulas were established in the case of branching Pearson random walks with exponentially distributed jumps. In this letter, we derive two key results: first, we show that such formulas strikingly carry over to the much broader class of branching processes with arbitrary jumps, and have thus a universal character; second, we obtain a stronger version of these formulas relating the travelled length density and the collision density at any point of the phase space. Our results are key to such technological issues as the analysis of radiation flow for nuclear reactor design and medical diagnosis and apply more broadly to physical and biological systems with diffusion, reproduction and death.