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# Contents

## 1 Introduction 1

1.1 Content of the paper and organization 3

1.2 Related works / Motivation 5

1.3 The nature of general dispersion models, and restriction to "valid ones" 7

1.4 Valid continuous dispersion models (valid CDM) : formal addendum 8

1.5 List of examples of valid continuous dispersion processes 10

1.6 Main universality results for valid CDM: deterministic masses case 14

1.7 Reduction to the Right Diffusion with Constant Speed and excursion sizes 19

1.8 Proof of Lemma 1.5, Lemma 1.6 and Lemma 1.11 20

1.9 Proof of Theorem 1.5($v$) 24

## 2 Valid discrete dispersion models 25

2.1 Main universality result for valid discrete dispersion models 26

2.2 Distribution of the number of CC $N_k^{(n)}$. 28

2.3 Examples of DDMs. 29

## 3 Asymptotics 30

3.1 Discussion regarding random masses results 30

3.2 Discrete masses: comparison between continuous and discrete spaces 30

3.3 Random models (and caravans) 35

## 4 Energy dissipation and other cost associated with dispersion model 39

4.1 Cost of parking construction for general unitary cost function 39

4.2 Proof of Lemma 4.4 44

4.3 Discussion: Cost for general diffusion processes 54
<table><tr><td>5 Appendix</td><td>55</td></tr><tr><td>5.1 Justification of fluid approximation of Section 1.5.6</td><td>55</td></tr><tr><td>5.2 Justification of (1.18)</td><td>55</td></tr><tr><td>5.3 Computation of E(&lt;x &rarr; x<sup>k</sup>, M<sub>&lambda;</sub>&gt;)</td><td>56</td></tr></table>