The two-type continuum Richardson model: Non-dependence of the survival of both types on the initial configuration

We consider the model of Deijfen et al. for the competing growth of two infection types in Rd, based on the Richardson model on Zd. Stochastic ball-shaped infection outbursts transmit the infection type of the center of the ball to all points of the ball that are not yet infected. Relevant parameters of the model are the initial infection configuration, the (typedependent) growth rates and the radius distribution of the infection outbursts. The main question is that of coexistence: For what values of the parameters is there a positive probability that both types grow unboundedly? It is known that for this question the initial configuration basically is irrelevant, provided certain technical assumptions on the radius distribution are satisfied. Here we show how to get rid of these assumptions, introducing a slight generalization of the model, where immune regions and delayed initial infection configurations are allowed.