  The evolution of several physical and biological systems, ranging from
neutron transport in multiplying media to epidemics or population dynamics, can
be described in terms of branching exponential flights, a stochastic process
which couples a Galton-Watson birth-death mechanism with random spatial
displacements. Within this context, one is often called to assess the length
$\ell_V$ that the process travels in a given region $V$ of the phase space, or
the number of visits $n_V$ to this same region. In this paper, we address this
issue by resorting to the Feynman-Kac formalism, which allows characterizing
the full distribution of $\ell_V$ and $n_V$ and in particular deriving explicit
moment formulas. Some other significant physical observables associated to
$\ell_V $ and $n_V$, such as the survival probability, are discussed as well,
and results are illustrated by revisiting the classical example of the rod
model in nuclear reactor physics.
