Universiteit Leiden

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Proefschrift

Dynamical Gibbs-non-Gibbs transitions: a study via coupling and large deviations

Promotores: F.H.J. Redig, W.T.F. den Hollander

Auteur
Feijia Wang
Datum
07 november 2012
Links
Thesis in Leiden Repository

In this thesis we use both the two-layer and the large-deviation approach to study the conservation and loss of the Gibbs property for both lattice and mean-field spin systems. Chapter 1 gives general backgrounds on Gibbs and non-Gibbs measures and outlines the the two-layer and the large-deviation approach. Chapter 2 studies the transforms of one-dimensional lattice spin systems. We start from a Gibbs measure with infinite range interaction and consider both deterministic and stochastic transformations K. Using the two-layer approach we prove that the constrained system has a unique Gibbs measure for every choice of transformed configuration, as long as the range of K is finite. This implies that the associated transformed Gibbs measures are always Gibbs. Further, we prove that if the initial interaction is exponentially decaying, then the transformed interaction decays exponentially as well, while if the initial interaction is polynomially decaying (with an exponent large enough so that the system is in the uniqueness regime), then the transformed interaction decays polynomially as well (with a smaller power). The proofs of these results use the house-of-cards coupling argument. Chapters 3 and 4 provide new and explicitly computable examples of Gibbs-non-Gibbs transitions by using the large-deviation approach. These examples include independent Brownian motions, Ornstein-Uhlenbeck processes, and birth-death processes. Chapter 4 computes the Feng-Kurtz Hamiltonian and Lagrangian associated to the large deviations of the trajectory of the empirical distribution for independent Markov processes, and of the empirical measure for translation invariant interacting Markov processes. We treat both the case of jump processes (continuous-time Markov chains and interacting particle systems) and the case of diffusion processes. For di usion processes, the Lagrangian is a quadratic form of the deviation of the trajectory from the Kolmogorov forward equation. In all cases, the Lagrangian can be interpreted as a relative entropy (density) per unit time.