Discrete Feynman-Kac formulas for branching random walks

TL;DR

This paper derives discrete Feynman-Kac formulas for branching random walks, enabling the calculation of visit probabilities and moments, with connections to residence times in Markovian processes.

Contribution

It introduces discrete Feynman-Kac equations for branching random walks, extending existing formulas to a broader class of stochastic processes.

Findings

Derived formulas for probability and moments of visits in branching random walks

Connected discrete Feynman-Kac formulas to residence times in the diffusion limit

Applicable to physical and biological systems like neutron multiplication and population dynamics

Abstract

Branching random walks are key to the description of several physical and biological systems, such as neutron multiplication, genetics and population dynamics. For a broad class of such processes, in this Letter we derive the discrete Feynman-Kac equations for the probability and the moments of the number of visits $n_V$ of the walker to a given region $V$ in the phase space. Feynman-Kac formulas for the residence times of Markovian processes are recovered in the diffusion limit.