Results on Branching Random Walks and Rumor Processes via Germ Order

Germ order is a non-standard stochastic order defined through the comparison of the generating functions of the processes. This order was first introduced for branching random walks with a constant breeding law and independent dispersal of offspring, which are characterized by a one-dimensional generating function. In this work, we investigate the properties of the extension of this concept to processes characterized by a multidimensional generating function, such as, general branching random walks and rumor processes. In particular, we use germ ordering to characterize the behavior of certain branching random walks and rumor processes with inhomogeneous breeding/transmitting laws.