The Q-process for the general mother-dependent multitype branching process
The general mother-dependent neutral mutations model is a multitype branching process describing a population evolving in discrete time, in which mutations occur among a finite set of alleles: each mutant, independently of the others, acquires an allele different from its mother's, according to a prescribed distribution. We study the associated mutant and clone-mutant chains conditioned on extinction in the distant future, that is, their $Q$-processes, and we give an explicit description of their laws in the (sub)critical and in the supercritical regimes alike. Even under this conditioning, the clone-mutant process associated with the mother-dependent neutral mutations model is obtained from the corresponding infinite-alleles BGW process (studied in Bertoin (2010) and Blancas and Rivero (2018)) by an independent multinomial colouring with specific parameters. We also obtain the spine decomposition of the multitype mutant chain, from which we derive a Kesten--Stigum theorem in the supercritical regime and a genealogical construction of the Q-process in the (sub)critical one.