A Widom-Rowlinson Jump Dynamics in the Continuum

We study the dynamics of an infinite system of point particles of two types. They perform random jumps in $\mathbf{R}^d$ in the course of which particles of different types repel each other whereas those of the same type do not interact. The states of the system are probability measures on the corresponding configuration space, the global (in time) evolution of which is constructed by means of correlation functions. It is proved that for each initial sub-Poissonian state $\mu_0$, the states evolve $\mu_0 \mapsto \mu_t$ in such a way that $\mu_t$ is sub-Poissonian for all $t>0$. The mesoscopic (approximate) description of the evolution of states is also given. The stability of translation invariant stationary states is studied. In particular, we show that some of such states can be unstable with respect to space-dependent perturbations.