Properties of branching exponential flights in bounded domains

TL;DR

This paper investigates the properties of branching exponential flights within bounded domains, establishing relations between boundary and interior starting points, and deriving key physical observables using the Feynman-Kac formalism.

Contribution

It introduces a novel framework linking trajectory properties in bounded regions to boundary conditions and applies the Feynman-Kac formalism to derive physical observables.

Findings

Total length and collision counts can be computed using Feynman-Kac.

Survival and escape probabilities are explicitly derived.

Relations between boundary and interior starting points are established.

Abstract

Branching random flights are key to describing the evolution of many physical and biological systems, ranging from neutron multiplication to gene mutations. When their paths evolve in bounded regions, we establish a relation between the properties of trajectories starting on the boundary and those starting inside the domain. Within this context, we show that the total length travelled by the walker and the number of performed collisions in bounded volumes can be assessed by resorting to the Feynman-Kac formalism. Other physical observables related to the branching trajectories, such as the survival and escape probability, are derived as well.