Dynamic Boolean models

Consider an ordinary Boolean model, that is, a homogeneous Poisson point process in Rd, where the points are all centres of random balls with i.i.d. radii. Now let these points move around according to i.i.d. stochastic processes. It is not hard to show that at each fixed time t we again have a Boolean model with the original distribution. Hence if the original model is supercritical then, for any t, the probability of having an unbounded occupied component at time t equals 1. We show that under mild conditions on the dynamics (e.g. for Brownian motion) we can interchange the quantifiers in the above statement, namely: if the original model is supercritical, then the probability of having an unbounded occupied component for all t simultaneously equals 1. Roughly analogous statements are valid for the subcritical regime, under some further mild conditions.