The demographic variation process of multitype branching random fields

A system of particles of k types in Rd is considered, where each particle, depending on its type, migrates and lives a random amount of time, at the end of which it branches according to a multitype law. The demographic variation process is a non-Markovian process which measures the changes in the system due to the branching. The asymptotic fluctuations of the demographic variation are studied for three different rescalings, the limit fluctuation processes being generalized Gaussian. The main objective is to identify when the limit fluctuation processes of the demographic variation are Markovian. It is shown that the Markov property holds only in some cases of critical branching, which depend on the type of rescaling.