### Abstract

We study the problem of relating the long time behavior of finite and infinite systems of locally interacting components. We consider in detail a class of lincarly interacting diffusions x(t)={x_{ i} (t), i ∈ ℤ^{ d} } in the regime where there is a one-parameter family of nontrivial invariant measures. For these systems there are naturally defined corresponding finite systems, {Mathematical expression}, with {Mathematical expression}. Our main result gives a comparison between the laws of x(t_{ N} ) and x^{ N} (t_{ N} ) for times t_{ N} →∞ as N→∞. The comparison involves certain mixtures of the invariant measures for the infinite system.

Original language | English (US) |
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Pages (from-to) | 165-197 |

Number of pages | 33 |

Journal | Probability Theory and Related Fields |

Volume | 103 |

Issue number | 2 |

DOIs | |

State | Published - Jun 1 1995 |

### Keywords

- Mathematics Subject Classification (1991): 60K35

### ASJC Scopus subject areas

- Analysis
- Statistics and Probability
- Statistics, Probability and Uncertainty

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## Cite this

Cox, J. T., Greven, A., & Shiga, T. (1995). Finite and infinite systems of interacting diffusions.

*Probability Theory and Related Fields*,*103*(2), 165-197. https://doi.org/10.1007/BF01204213