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**Asymptotics of particle trajectories in infinite one-dimensional systems with collisions.**
*(English)*
Zbl 0578.60094

Summary: We investigate the asymptotic behavior of the trajectories of a tagged particle (tp) in an infinite one-dimensional system of point particles. The particles move independently when not in contact: the only interactions are Harris type generalized elastic collisions which prevent crossings. This is achieved by relabeling the independent trajectories when they cross. When these trajectories are differentiable, as in particles with velocities undergoing Ornstein-Uhlenbeck processes, collisions correspond to exchange of velocities.

We prove very generally that the suitably scaled tp trajectory converges (weakly) to a simple Gaussian process. This extends the results of F. Spitzer [J. Math. Mech. 18, 973-989 (1969; Zbl 0184.211)] for Newtonian particles to very general non-crossing processes. The proof is based on the consideration of the simpler process which counts the crossings of the origin by the independent trajectories.

We prove very generally that the suitably scaled tp trajectory converges (weakly) to a simple Gaussian process. This extends the results of F. Spitzer [J. Math. Mech. 18, 973-989 (1969; Zbl 0184.211)] for Newtonian particles to very general non-crossing processes. The proof is based on the consideration of the simpler process which counts the crossings of the origin by the independent trajectories.

### MSC:

60K35 | Interacting random processes; statistical mechanics type models; percolation theory |

60J70 | Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) |

### Citations:

Zbl 0184.211
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\textit{D. Dürr} et al., Commun. Pure Appl. Math. 38, 573--597 (1985; Zbl 0578.60094)

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### References:

[1] | Harris, J. Appl. Prob. 2 pp 323– (1965) |

[2] | Spitzer, J. Math. Mech. 18 pp 973– (1968) |

[3] | Gisselquist, Annals of Prob. 1 pp 231– (1973) · Zbl 0263.60047 |

[4] | Stochastic Processes, Wiley, New York, 1953. |

[5] | Convergence of Probability Measures, Wiley, New York, 1968. · Zbl 0172.21201 |

[6] | preprint. |

[7] | Stochastic Processes and the Wiener Integral, M. Dekker, New York, 1973. |

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