Summary
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 diffusionsx(t)={x i (t),i ∈ ℤd} in the regime where there is a one-parameter family of nontrivial invariant measures. For these systems there are naturally defined corresponding finite systems,\(x^N (t) = \left\{ {x_i^N (t),i \in \Lambda _N } \right\}\), with\(\Lambda _N = ( - N,N]^d \cap \mathbb{Z}^d\). Our main result gives a comparison between the laws ofx(t N ) andx N(t N ) for timest N →∞ asN→∞. The comparison involves certain mixtures of the invariant measures for the infinite system.
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Partly supported by the U.S. Army Research Office through the Mathematical Sciences Institute of Cornell University, by the National Science Foundation, and by the National Security Agency
Research supported in part by the DFG
Partly supported by S.R.63540155 of Japan Ministry of Education
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Cox, J.T., Greven, A. & Shiga, T. Finite and infinite systems of interacting diffusions. Probab. Th. Rel. Fields 103, 165–197 (1995). https://doi.org/10.1007/BF01204213
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DOI: https://doi.org/10.1007/BF01204213