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The handle http://hdl.handle.net/1887/20110 holds various files of this Leiden University dissertation.
Author: Wang, Feijia
Title: Dynamical Gibbs-non-Gibbs transitions : a study via coupling and large deviations Issue Date: 2012-11-07
Dynamical Gibbs-non-Gibbs transitions:
a study via coupling and large deviations
Proefschrift
ter verkrijging van
de graad van Doctor aan de Universiteit Leiden,
op gezag van Rector Magnificus prof. mr. P.F. van der Heijden, volgens besluit van het College voor Promoties
te verdedigen op woensdag 7 november 2012 klokke 10:00 uur
door
Feijia Wang
geboren te Guangxi
Promotores: Prof. dr. F.H.J. Redig
Prof. dr. W.Th.F. den Hollander
Beoordelingscommissie: Prof. dr. A.C.D. van Enter Prof. dr. C. K¨ulske
Dr. A. Le Ny
Acknowledgements
I would like to thank people who have been instrumental in helping this PhD- project come to a good end.
I would like to thank Prof. Frank Redig and Prof. Frank den Hollander for accepting me as a PhD student in Leiden. I would like to thank Prof. Redig for his teachings, his questions, his answers, his guidance, etc: I owe him a lot. I would like to thank Prof. den Hollander for letting me be part of the probability theory group and for his additional guidance. I would also like to thank Dr. Alex Opoku for answering many technical questions and for helping me to improve the introduction to my thesis. His course on Gibbs measures was very impressive. I would further like to thank the members of the reading committee for their critical remarks. Finally, I would like to thank my colleagues in the probability theory group of the institute for their friendliness and the Mathematisch Instituut of Leiden University for its financial support.
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Contents
1 Introduction 1
1.1 Overview of thesis . . . . 4
1.1.1 Conservation of Gibbsianness for lattice spin systems . . . . 4
1.1.2 Dynamical Gibbs-non-Gibbs transitions for mean-field spin systems 5 1.1.3 Large deviations for the trajectory of the empirical distribution and empirical measure . . . . 8
1.2 Generalities on Gibbs and non-Gibbs measures . . . . 8
1.2.1 Gibbs measures . . . . 8
1.2.2 Transformation of Gibbs measures and phenomenon of non-Gibbsianness 12 1.2.3 The two-layer picture . . . . 13
1.3 Large deviation theory and optimal solutions for rate function . . . . 15
1.3.1 Large deviation principle . . . . 15
1.3.2 Large deviations for stochastic processes . . . . 18
1.3.3 The Feng-Kurtz scheme . . . . 19
1.3.4 Uniqueness and non-uniqueness of optimal trajectories . . . . 20
2 Transformations of one-dimensional Gibbs measures with infinite-range interaction 27 2.1 Introduction . . . . 27
2.2 Gibbs measures and their transformations . . . . 28
2.2.1 One-dimensional Gibbs measures . . . . 28
2.2.2 Transformations of Gibbs measures . . . . 30
2.3 Stochastic single-site transformations . . . . 31
2.3.1 The transformed potential . . . . 36
2.4 Deterministic single-site transformations . . . . 37
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CONTENTS
2.4.1 Exponentially decaying potential . . . . 39
2.4.2 Power-law decaying potential . . . . 41
2.5 Finite-block transformations . . . . 42
3 Gibbs-non-Gibbs transitions via large deviations: computable examples 45 3.1 Introduction . . . . 45
3.2 The Feng-Kurtz scheme, Euler-Lagrange trajectories, bad configurations . . 46
3.3 Diffusion processes with small variance conditioned on the future . . . . 48
3.3.1 Brownian motion . . . . 49
3.3.2 Brownian motion with constant drift . . . . 52
3.3.3 Other rate functions for the initial measure and corresponding behav- ior of Brownian motion . . . . 52
3.4 The Ornstein-Uhlenbeck process . . . . 56
3.4.1 The Ornstein-Uhlenbeck process with constant external field . . . . 57
3.4.2 General drift. . . . 58
3.5 Approximately deterministic walks in d = 1 . . . . 60
3.5.1 Constant birth and death rates . . . . 62
3.5.2 Mean-field independent spin flips . . . . 62
3.5.3 Independent spin-flips in a field . . . . 64
4 Large deviations for the trajectory of the empirical distribution and em- pirical measure 69 4.1 Introduction . . . . 69
4.2 The trajectory of the empirical distribution: general case . . . . 71
4.3 Finite-state space continuous-time Markov chains . . . . 73
4.3.1 Hamiltonian trajectories for finite Markov chains . . . . 75
4.3.1.1 Two-state symmetric flipping . . . . 77
4.4 Diffusion processes . . . . 78
4.5 Trajectory of the empirical measure . . . . 81
4.5.1 Context and notation . . . . 81
4.5.2 Translation invariant sequence of local generators . . . . 82
4.5.3 Trajectory of the empirical measure . . . . 83
4.5.4 The Feng-Kurtz Hamiltonian . . . . 85
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CONTENTS
4.5.5 Interacting particle systems . . . . 86 4.5.6 Diffusion processes: computation of the Lagrangian. . . . 88
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CONTENTS
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