E. Barkai, E. Aghion, and D. A. Kessler, From the Area under the Bessel Excursion to Anomalous Diffusion of Cold Atoms, Physical Review X, vol.19, issue.2, p.21036, 2014.
DOI : 10.1007/BF00533253

N. B. Abdallah, A. Mellet, and M. , ANOMALOUS DIFFUSION LIMIT FOR KINETIC EQUATIONS WITH DEGENERATE COLLISION FREQUENCY, Mathematical Models and Methods in Applied Sciences, vol.49, issue.11, pp.2249-2262, 2011.
DOI : 10.1063/1.1666510

N. B. Abdallah, A. Mellet, and M. , Fractional diffusion limit for collisional kinetic equations: a Hilbert expansion approach, Kinet. Relat. Models, vol.4, pp.873-900, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00807757

A. Bensoussan, J. L. Lions, and G. Papanicolaou, Boundary layers and homogenization of transport processes, Publications of the Research Institute for Mathematical Sciences, vol.15, issue.1, pp.53-157, 1979.
DOI : 10.2977/prims/1195188427
URL : http://www.ems-ph.org/journals/show_pdf.php?issn=0034-5318&vol=15&iss=1&rank=3

P. Biane and M. Yor, Valeurs principales associées aux temps locaux browniens, Bull. Sci. Math, vol.111, pp.23-101, 1987.

T. Bodineau, I. Gallagher, and L. Saint-raymond, The Brownian motion as the limit of a deterministic system of hard-spheres, Inventiones mathematicae, vol.18, issue.2, pp.493-553, 2016.
DOI : 10.1007/BF01239014
URL : https://hal.archives-ouvertes.fr/hal-01137218

Y. Castin, J. Dalibard, and C. Cohen-tannoudji, The limits of Sisyphus cooling in Light Induced Kinetic Effects on Atoms, Ions and Molecules, Proceedings of the workshop " Light Induced Kinetic Effects on Atom, Ions and Molecules, 1990.

P. Cattiaux, D. Chaffa¨?chaffa¨?, and A. Guillin, Central limit theorems for additive functionals of ergodic Markov diffusions processes, ALEA, Lat. Am. J. Probab. Math. Stat, vol.9, pp.337-382, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00585271

P. Cattiaux, N. Gozlan, A. Guillin, and C. Roberto, Functional Inequalities for Heavy Tailed Distributions and Application to Isoperimetry, Electronic Journal of Probability, vol.15, issue.0, pp.346-385, 2010.
DOI : 10.1214/EJP.v15-754
URL : https://hal.archives-ouvertes.fr/hal-00666780

P. Cattiaux, E. Nasreddine, and M. , Diffusion limit for kinetic Fokker-Planck equation with heavy tail equilibria : the critical case

C. Donati-martin, B. Roynette, P. Vallois, and M. Yor, < 1, Studia Scientiarum Mathematicarum Hungarica, vol.45, issue.2, pp.45-207, 2008.
DOI : 10.1556/SScMath.2007.1033

O. Hirschbeg, D. Mukamel, and G. M. Schütz, Diffusion in a logarithmic potential: scaling and selection in the approach to equilibrium, J. Stat. Mech.: Theory and Experiments, 2012.

K. Itô and H. P. Mckean, Diffusion processes and their sample paths, 1965.

J. Jacod and A. N. Shiryaev, Limit theorems for stochastic processes, 2003.
DOI : 10.1007/978-3-662-02514-7

T. Jeulin and M. Yor, Sur les distributions de certaines fonctionnelles du mouvement Brownien, pp.210-226, 1981.
DOI : 10.1007/BFb0070884

O. Kallenberg, Foundations of modern probability, 2002.
DOI : 10.1007/978-1-4757-4015-8

P. Langevin, Sur la théorie du mouvement brownien, C. R. Acad. Sci, vol.146, pp.530-533, 1908.

E. W. Larsen and J. B. Keller, Asymptotic solution of neutron transport problems for small mean free paths, Journal of Mathematical Physics, vol.15, issue.1, pp.75-81, 1974.
DOI : 10.13182/NSE61-1

G. Lebeau and M. , Diffusion approximation for Fokker Planck with heavy tail equilibria : a spectral method in dimension 1

K. Marksteiner, P. Ellinger, and . Zoller, Anomalous diffusion and L??vy walks in optical lattices, Physical Review A, vol.49, issue.5, p.3409, 1996.
DOI : 10.1103/PhysRevA.49.R4297

A. Mellet, Fractional diffusion limit for collisional kinetic equations: A moments method, Indiana University Mathematics Journal, vol.59, issue.4, pp.1333-1360, 2010.
DOI : 10.1512/iumj.2010.59.4128
URL : http://arxiv.org/pdf/0910.1570

A. Mellet, S. Mishler, and C. Mouhot, Fractional Diffusion Limit for Collisional Kinetic Equations, Archive for Rational Mechanics and Analysis, vol.346, issue.2, pp.199-493, 2011.
DOI : 10.1016/j.physa.2004.08.006
URL : https://hal.archives-ouvertes.fr/hal-00321478

J. Milton, T. Komorowski, and S. Olla, Limit theorems for additive functionals of a Markov chain, Ann. Appl. Probab, vol.19, pp.2270-2300, 2009.

E. Nasreddine and M. , Diffusion limit of Fokker???Planck equation with heavy tail equilibria, ESAIM: Mathematical Modelling and Numerical Analysis, vol.49, issue.1, pp.1-17, 2015.
DOI : 10.1007/s00205-010-0354-2

D. Revuz and M. Yor, Continuous martingales and Brownian motion. Third Edition, 2005.

Y. Sagi, M. Brook, I. Almog, and N. Davidson, Observation of Anomalous Diffusion and Fractional Self-Similarity in One Dimension, Physical Review Letters, vol.108, issue.9, p.93002, 2012.
DOI : 10.1103/PhysRevE.84.041111