Discrete Feynman-Kac formulas for branching random walks

Branching random walks are key to the description of several physical and biological systems, such as neutron transport in multiplying media, epidemics and population genetics. Within this context, assessing the number of visits n_V of the walker to a given region V in the phase space plays a fundamental role. In this letter we derive the discrete Feynman-Kac equations for the distribution of n_V as a function of the starting point of the walker, when the process is observed up to the n-th generation. We provide also the recurrence relation for the moments of the distribution, and illustrate this formalism on a problem in reactor physics. Feynman-Kac formulas for the residence times of Markovian processes are recovered in the diffusion limit.