Properties of branching exponential flights in bounded domains

CEA/Saclay, DEN/DANS/DM2S/SERMA/LTSD - Gif-sur-Yvette, France

Received: 27 July 2012
Accepted: 5 November 2012

Abstract

In a series of recent works, important results have been reported concerning the statistical properties of exponential flights evolving in bounded domains, a widely adopted model for finite-speed transport phenomena (Blanco S. and Fournier R., Europhys. Lett., 61 (2003) 168; Mazzolo A., Europhys. Lett., 68 (2004) 350; Bénichou O. et al., Europhys. Lett., 70 (2005) 42). Motivated by physical and biological systems where random spatial displacements are coupled with Galton-Watson birth-death mechanisms, such as neutron multiplication, diffusion of reproducing bacteria or spread of epidemics, in this letter we extend those results in two directions, via a Feynman-Kac formalism. First, we characterize the occupation statistics of exponential flights in the presence of absorption and branching, and give explicit moment formulas for the total length travelled by the walker and the number of performed collisions in a given domain. Then, we show that the survival and escape probability can be derived as well by resorting to a similar approach.