### Abstract

Consider a distinguished, or tagged particle in zero-range dynamics on Z^{d} with rate g whose finite-range jump probabilities p possess a drift ∑ j p (j) ≠ 0. We show, in equilibrium, that the variance of the tagged particle position at time t is at least order t in all d ≥ 1, and at most order t in d = 1 and d ≥ 3 for a wide class of rates g. Also, in d = 1, when the jump distribution p is totally asymmetric and nearest-neighbor, and the rate g (k) increases, and g (k) / k either decreases or increases with k, we show the diffusively scaled centered tagged particle position converges to a Brownian motion with a homogenized diffusion coefficient in the sense of finite-dimensional distributions. Some characterizations of the tagged particle variance are also given.

Original language | English (US) |
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Pages (from-to) | 215-232 |

Number of pages | 18 |

Journal | Annales de l'institut Henri Poincare (B) Probability and Statistics |

Volume | 43 |

Issue number | 2 |

DOIs | |

State | Published - Mar 1 2007 |

Externally published | Yes |

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### Keywords

- Diffusive
- Invariance principle
- Tagged particle
- Variance
- Zero-range

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty