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Time scales in large systems of Brownian particles with stochastic synchronization

We consider a system $x(t)=(x_{1}(t),...,x_{N}(t))$ consisting of $N$ Brownian particles with synchronizing interaction between them occurring at random time moments $\{τ_{n}\}_{n=1}^{\infty}$. Under assumption that the free Brownian motions and the sequence $\{τ_{n}\}_{n=1}^{\infty}$ are independent we study asymptotic properties of the system when both the dimension~$N$ and the time~$t$ go to infinity. We find three time scales $t=t(N)$ of qualitatively different behavior of the system.