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Gibbs-non-Gibbs dynamical transitions for mean-field interacting Brownian motions

Author

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  • den Hollander, F.
  • Redig, F.
  • van Zuijlen, W.

Abstract

We consider a system of real-valued spins interacting with each other through a mean-field Hamiltonian that depends on the empirical magnetisation of the spins. The system is subjected to a stochastic dynamics where the spins perform independent Brownian motions. Using large deviation theory we show that there exists an explicitly computable crossover time tc∈[0,∞] from Gibbs to non-Gibbs. We give examples of immediate loss of Gibbsianness (tc=0), short-time conservation and large-time loss of Gibbsianness (tc∈(0,∞)), and preservation of Gibbsianness (tc=∞). Depending on the potential, the system can be Gibbs or non-Gibbs at the crossover time t=tc.

Suggested Citation

  • den Hollander, F. & Redig, F. & van Zuijlen, W., 2015. "Gibbs-non-Gibbs dynamical transitions for mean-field interacting Brownian motions," Stochastic Processes and their Applications, Elsevier, vol. 125(1), pages 371-400.
    • Handle: RePEc:eee:spapps:v:125:y:2015:i:1:p:371-400
    DOI: 10.1016/j.spa.2014.09.011
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    Cited by:

    1. W. Zuijlen, 2018. "Large Deviations of Continuous Regular Conditional Probabilities," Journal of Theoretical Probability, Springer, vol. 31(2), pages 1058-1096, June.
    2. Dommers, S. & Külske, C. & Schriever, P., 2017. "Continuous spin models on annealed generalized random graphs," Stochastic Processes and their Applications, Elsevier, vol. 127(11), pages 3719-3753.

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