Evolution of States of a Continuum Jump Model with Attraction

2016/12/24 by Kozitsky, Yuri
#34G10 #47D06 #60J80 #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.1612.08198

Abstract

We study a model of an infinite system of point particles in \mathdsR^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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