Abstract
We study a model of an infinite system of point particles in \(\mathbb {R}^d\) performing random jumps with attraction. The system’s states are probability measures on the space of particle configurations, and their evolution is described by means of Kolmogorov and Fokker-Planck equations. Instead of solving these equations directly we deal with correlation functions evolving according to a hierarchical chain of differential equations, derived from the Kolmogorov equation. Under quite natural conditions imposed on the jump kernels—and analyzed in the paper—we prove that this chain has a unique classical sub-Poissonian solution on a bounded time interval. This gives a partial answer to the question whether the sub-Poissonicity is consistent with any kind of attraction. We also discuss possibilities to get a complete answer to this question.
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Acknowledgements
The present research was supported by the European Commission under the project STREVCOMS PIRSES-2013-612669.
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Kozitsky, Y. (2018). Evolution of States of a Continuum Jump Model with Attraction. In: Agranovsky, M., Golberg, A., Jacobzon, F., Shoikhet, D., Zalcman, L. (eds) Complex Analysis and Dynamical Systems. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-70154-7_10
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DOI: https://doi.org/10.1007/978-3-319-70154-7_10
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