Kalikow decomposition for counting processes with stochastic intensity *

We propose a new Kalikow decomposition and the corresponding Perfect Simulation algorithm for continuous time multivariate counting processes, on potentially infinite networks. We prove the existence of such a decomposition in various cases. This decomposition is not unique and we discuss the choice of the decomposition in terms of algorithmic efficiency. We apply these methods on several examples: linear Hawkes process, age dependent Hawkes process, exponential Hawkes process.