# Contraction of the Wasserstein distance between two solutions

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## Two useful auxiliary considerations

In this section, we leave the study of the ergodicity of the billiard process apart and establish auxiliary results regarding two objects:
• the evolution of a single particle in contact with thermal baths in Subsection 3.4.1.
• the evolution of the unfolded version of the billiard process in Subsection 3.4.2.

### Marginal action of the thermal baths on a single particle

In order to quantify the marginal action of the thermal baths on the molecules, this subsection addresses the motion of a single molecule in the interval [0, b], with b > 0. The molecule travels at constant velocity in the interior of the interval. When it reaches the leftmost boundary, it is reflected with a random velocity drawn according to the density φi+,+(p) on [0,+∞), with i+ ∈ {1, 2}. When it reaches the rightmost boundary, it is reflected with a random velocity, drawn according to the density φi−,−(−p) on (−∞, 0], with i− ∈ {1, 2}. The position of the molecule is denoted by q(t) ∈ [0, b], and its velocity is denoted by p(t) ∈ R, see Figure 3.3.
The time taken by the particle to cross the interval after a velocity update is distributed according to the densities ψb;i+,+(r) and ψb;i−,−(r), defined as follows.

#### Unfolding the trajectories

A classical trick in the study of deterministic polygonal billiards is the unfolding procedure, which can roughly be described as follows: when the billiard particle hits a boundary of the polygon, instead of reflecting the trajectory specularly, we rather reflect the polygon with respect to the boundary and let the billiard particle keep on moving in straight line. Repeating the procedure at each reflection, the trajectory of the billiard particle becomes a half line, evolving across a sequence of iterated copies of the original billiard table. We refer to the textbook by Tabachnikov  or the review article by Gutkin  for more precisions regarding the unfolding of polygonal billiards.
Clearly, the unfolding procedure is well adapted to specular reflections, but not to stochastic reflections. Therefore, we use a partial unfolding procedure and define the unfolded billiard process as follows.

The sequence of observation times

In this section, we explain how to construct a subset Y ⊂ Xnd of and a function τobs : Y → (0,+∞) satisfying the stability condition of Definition 3.3.8 and such that the associated Markov renewal process (Yn, τn)n≥0 hopefully satisfies the assumptions of the Markov Renewal Theorem 3.3.7. The definition of the sequence of observation times is detailed in Subsection 3.5.1. A precise description of the Markov renewal process thus obtained is made in Subsection 3.5.2. We finally give a few properties of the stationary distributions of the Markov chain in Subsection 3.5.3.

On the stationary distributions of (Yn)n0

Recall that stationary σ-finite distributions for the Markov chain (Yn)n≥0 were introduced in Definition 3.3.2. Without addressing the issue of the existence such a distribution yet, we already derive necessary conditions. For all (i, ǫ, ⋆) ∈ , let us denote by ¯ν|Yi,ǫ ⋆ the restriction of a stationary σ-finite distribution ¯ν to the elementary section Yi,ǫ ⋆ .

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Marginal distribution of the normal component of the velocity

The following remark on the shape of ¯ν on the boundary sections can immediately be formulated. Lemma 3.5.8. For all stationary σ-finite distribution ¯ν, for all (i, ǫ) ∈ {1, 2}×{+,−}, there exist positive and σ-finite measures ¯ν1 |Yi,ǫ bo (dq1dp1) and ¯ν2 |Yi,ǫ bo (dq2dp2) such that ¯ν|Yi,ǫ bo = ¯ν1 |Yi,ǫ bo (dq1dp1) ⊗ ¯ν2 |Yi,ǫ bo (dq2dp2).

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

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