  Many phenomena can be modeled as network dynamics with punctuate
interactions. However, most relevant dynamics do not allow for computational
tractability. To circumvent this difficulty, the Poisson Hypothesis regime
replaces interaction times between nodes by independent Poisson processes,
allowing for tractability in several cases, such as intensity-based models from
computational neuroscience. This hypothesis is usually only conjectured, or
numerically validated. In this work, we introduce a class of processes in
continuous time called continuous fragmentation-interaction-aggregation
processes, by analogy with previously introduced processes in discrete time.
The state of each node, described by the stochastic intensity of an associated
point process, aggregates arrivals from its neighbors and is fragmented upon
departure. We consider the replica-mean-field version of such a process, which
is a physical system consisting of randomly interacting copies of the network
of interest. Generalizing results proved in discrete time and in the particular
case of excitatory intensity-based neural dynamics, we prove that the Poisson
Hypothesis holds at the limit of an infinite number of replicas.
