Marginal action of the thermal baths on a single particle

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Table of contents

1 Introduction 
1.1 Introduction de la Partie I
1.2 Introduction de la Partie II
1.3 Introduction de la Partie III
I Le modèle d’échange complet
2 Thermodynamique du modèle d’échange complet 
2.1 Introduction
2.2 The deterministic CEM
2.3 The thermalised CEM
2.4 Conclusions
2.A Derivation of the expression of Lf , Ldr and Li
2.B Computation of pairwise marginal statistics
3 États stationnaires hors de l’équilibre du modèle d’échange complet 
3.1 Introduction
3.2 Model and results
3.3 General sketch of the proof
3.4 Two useful auxiliary considerations
3.5 The sequence of observation times
3.6 Proof of Theorem 3.2.5
3.A A formal construction of the billiard process
3.B Recurrence of the set
II Systèmes de particules interagissant à travers leur rang 
4 Propagation du chaos et comportement en temps long 
4.1 Introduction
4.2 Probabilistic approximation of the solution
4.3 Contraction of the Wasserstein distance between two solutions
4.4 Convergence to equilibrium
4.A Proof of Proposition 4.2.2
4.B Proof of Proposition 4.3.4
5 Grands systèmes stationnaires de particules 
5.1 Introduction
5.2 Stationary distributions
5.3 Expression and convergence of the Laplace transforms
5.4 Completion of the proof of Theorem 5.1.2
6 Le modèle d’Atlas en champ moyen 145
6.1 Introduction
6.2 The mean-field Atlas model
6.3 The weighted capital measure
6.4 Capital distribution curves
6.5 Performance of diversity weighted portfolios
6.A Proof of Proposition 6.3.1
6.B Long time behaviour of the asymptotic capital measure
III La dynamique des particules collantes multitype 
7 Interprétation probabiliste de systèmes paraboliques 
7.1 Introduction
7.2 Mild solutions to the system of Fokker-Planck equations
7.3 The nonlinear martingale problem
7.4 The multitype rank-based particle system
8 Limite petit bruit de processus de diffusion interagissant à travers leur ordre 
8.1 Introduction
8.2 The two-particle case
8.3 The rank-based case
8.4 The order-based case
8.5 Conclusion
8.A Proofs in the two-particle case
9 Dynamique des particules collantes multitype et systèmes hyperboliques 
9.1 Introduction
9.2 Main definitions and results
9.3 The Multitype Sticky Particle Dynamics
9.4 The uniform Lp stability estimates
9.5 Construction of solutions
9.A Proofs of technical results
Bibliographie