# American Institute of Mathematical Sciences

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February  2020, 40(2): 725-752. doi: 10.3934/dcds.2020059

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

 Instytut Matematyki, Uniwersytet Marii Curie-Skłodowskiej, Plac Marii Curie-Skłodowskiej 1, 20-031 Lublin, Poland

* Corresponding author

Received  July 2018 Revised  August 2019 Published  November 2019

Fund Project: The first author is supported by NCN grant 2017/25/B/ST1/00051

The dynamics is studied of an infinite collection of point particles placed in $\mathbb{R}^d$, $d\geq 1$. 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 $\mu$ by their correlation functions $k_\mu$ and is done in two steps: (a) constructing an evolution $k_{\mu_0} \to k_t$; (b) proving that $k_t$ is the correlation function of a unique sub-Poissonian state $\mu_t$.

Citation: Yuri Kozitsky, Krzysztof Pilorz. Random jumps and coalescence in the continuum: Evolution of states of an infinite particle system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (2) : 725-752. doi: 10.3934/dcds.2020059
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