Random jumps and coalescence in the continuum: Evolution of states of an infinite particle system

Abstract

The dynamics is studied of an infinite collection of point particles placed in \begin{document}$ \mathbb{R}^d $\end{document} , \begin{document}$ d\geq 1 $\end{document} . The particles perform random jumps with mutual repulsion accompanied by random merging of pairs of particles. The states of the collection are probability measures on the corresponding configuration space. The main result is the proof of the existence of the Markov evolution of states for a bounded time horizon if the initial state is a sub-Poissonian measure. The proof is based on representing sub-Poissonian measures \begin{document}$ \mu $\end{document} by their correlation functions \begin{document}$ k_\mu $\end{document} and is done in two steps: (a) constructing an evolution \begin{document}$ k_{\mu_0} \to k_t $\end{document} ; (b) proving that \begin{document}$ k_t $\end{document} is the correlation function of a unique sub-Poissonian state \begin{document}$ \mu_t $\end{document} .