The demographic variation process of branching random fields

A branching random field with immigration is considered. The demographic variation process is a non-Markovian signed measure-valued process which measures the changes in the system due to branchings and deaths in the population. The asymptotic behavior of this process under various scalings is studied. It is shown that the fluctuation limits are generalized Gaussian processes which are Markovian if and only if the branching is critical, in which case they are non-stationary generalized Ornstein-Uhlenbeck processes.