  We consider a linearized dynamical system modelling the flow rate of water
along the rivers and hillslopes of an arbitrary watershed. The system is
perturbed by a random rainfall in the form of a compound Poisson process. The
model describes the evolution, at daily time scales, of an interconnected
network of linear reservoirs and takes into account the differences in flow
celerity between hillslopes and streams as well as their spatial variation. The
resulting stochastic process is a piece-wise deterministic Markov process of
the Orstein-Uhlembeck type. We provide an explicit formula for the Laplace
transform of the invariant density of streamflow in terms of the geophysical
parameters of the river network and the statistical properties of the
precipitation field. As an application, we include novel formulas for the
invariant moments of the streamflow at the watershed's outlet, as well as the
asymptotic behavior of extreme discharge events.
