Convergences for a virus-like evolving population driven by mutually-exciting Hawkes processes

This paper presents a stochastic model motivated by the study of a virus-like evolving population with different mutation rates. This is a continuous time birth-death model: the birth processes are mutually-exciting Hawkes processes and the death process is also a Hawkes process. This structure for the births and the deaths does not allow, in general, to get the Markov property of the processes involved. But considering the couple given by the Hawkes processes and their intensities we are able to deduce the necessary and sufficient conditions for the Markov property of the couple. This property is the main tool to get the convergence results describing the behaviour of the population, and the existence of a phase transition at a critical fitness level.