D. Aregba-driollet, R. Natalini, and S. Tang, Explicit diffusive kinetic schemes for nonlinear degenerate parabolic systems, Mathematics of Computation, vol.73, issue.245, pp.63-94, 2004.
DOI : 10.1090/S0025-5718-03-01549-7

URL : https://hal.archives-ouvertes.fr/hal-00387865

P. Bak, How Nature Works: The science of Self-Organized Criticality, 1986.

V. Barbu, M. Röckner, and F. Russo, Probabilistic representation for solutions of an irregular porous media type equation: the degenerate case, Probability Theory and Related Fields, vol.34, issue.6
DOI : 10.1007/s00440-010-0291-x

URL : https://hal.archives-ouvertes.fr/inria-00410248

G. I. Barenblatt, On some unsteady motions of a liquid and gas in a porous medium, Akad. Nauk SSSR. Prikl. Mat. Meh, vol.16, pp.67-78, 1952.

S. Benachour, P. Chassaing, B. Roynette, and P. Vallois, Processus associésassociés`associésà l'´ equation des milieux poreux, Ann. Scuola Norm. Sup. Pisa Cl. Sci, vol.23, issue.4, pp.793-832, 1996.

P. Benilan, H. Brezis, and M. G. Crandall, A semilinear equation in L 1 (R N ), Ann. Scuola Norm, Sup. Pisa Cl. Sci, vol.2, issue.4, pp.523-555, 1975.

P. Benilan and M. G. Crandall, The continuous dependence on ? of solutions of u t ? ??(u) = 0, Indiana Univ, Math. J, vol.30, pp.161-177, 1981.

A. E. Berger, H. Brézis, and J. C. Rogers, A numerical method for solving the problem u t ? ?f (u) = 0, RAIRO Anal, Numér, vol.13, pp.297-312, 1979.

P. Blanchard, M. Röckner, and F. Russo, Probabilistic representation for solutions of an irregular porous media type equation, The Annals of Probability, vol.38, issue.5, pp.1870-1900, 2010.
DOI : 10.1214/10-AOP526

URL : https://hal.archives-ouvertes.fr/hal-00279975

M. Bossy and D. Talay, A stochastic particle method for some one-dimensional nonlinear p.d.e., Mathematics and Computers in Simulation, vol.38, issue.1-3, pp.43-50, 1995.
DOI : 10.1016/0378-4754(93)E0065-D

F. Bouchut, F. R. Guarguaglini, and R. Natalini, Diffusive BGK approximations for nonlinear multidimensional parabolic equations, Indiana University Mathematics Journal, vol.49, issue.2, pp.723-749, 2000.
DOI : 10.1512/iumj.2000.49.1811

A. W. Bowman, An alternative method of cross-validation for the smoothing of density estimates, Biometrika, vol.71, issue.2, pp.353-360, 1984.
DOI : 10.1093/biomet/71.2.353

H. Brezis and M. G. Crandall, Uniqueness of solutions of the initial-value problem for u t ? ??(u) = 0, J. Math. Pures Appl, vol.58, issue.9, pp.153-163, 1979.

R. Cafiero, V. Loreto, L. Pietronero, A. Vespignani, and S. Zapperi, Local Rigidity and Self-Organized Criticality for Avalanches, Europhysics Letters (EPL), vol.29, issue.2, pp.111-116, 1995.
DOI : 10.1209/0295-5075/29/2/001

P. Calderoni and M. Pulvirenti, Propagation of chaos for Burgers' equation, Ann. Inst. H. Poincaré Sect. A (N.S.), vol.39, pp.85-97, 1983.

F. Cavalli, G. Naldi, G. Puppo, and M. Semplice, High-Order Relaxation Schemes for Nonlinear Degenerate Diffusion Problems, SIAM Journal on Numerical Analysis, vol.45, issue.5, pp.2098-2119, 2007.
DOI : 10.1137/060664872

F. Cuvelier, Implementing Kernel Density Estimation on GPU: application to a probabilistic algorithm for PDEs of porous media type

J. M. Dawson, Particle simulation of plasmas, Rev. Modern Phys, pp.403-447, 1983.

A. Figalli and R. Philipowski, Convergence to the viscous porous medium equation and propagation of chaos, ALEA Lat, Am. J. Probab. Math. Stat, vol.4, pp.185-203, 2008.

E. Hairer, S. P. Nørsett, and G. Wanner, Solving ordinary differential equations. I, second ed, Series in Computational Mathematics, 1993.

A. Harten and S. Osher, Uniformly High-Order Accurate Nonoscillatory Schemes. I, SIAM Journal on Numerical Analysis, vol.24, issue.2, pp.279-309, 1987.
DOI : 10.1137/0724022

R. W. Hockney and J. W. Eastwood, Computer simulation using particles, 1981.

S. Jin and C. D. Levermore, Numerical Schemes for Hyperbolic Conservation Laws with Stiff Relaxation Terms, Journal of Computational Physics, vol.126, issue.2, pp.449-467, 1996.
DOI : 10.1006/jcph.1996.0149

S. Jin and Z. P. Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Communications on Pure and Applied Mathematics, vol.54, issue.3, pp.235-276, 1995.
DOI : 10.1002/cpa.3160480303

M. C. Jones, J. S. Marron, and S. J. Sheather, A Brief Survey of Bandwidth Selection for Density Estimation, Journal of the American Statistical Association, vol.53, issue.433, pp.91-401, 1996.
DOI : 10.1214/aoms/1177696810

B. Jourdain, Probabilistic approximation for a porous medium equation, Stochastic Process, Appl, vol.89, pp.81-99, 2000.

B. Jourdain and S. Méléard, Propagation of chaos and fluctuations for a moderate model with smooth initial data, Annales de l'Institut Henri Poincare (B) Probability and Statistics, vol.34, issue.6, pp.727-766, 1998.
DOI : 10.1016/S0246-0203(99)80002-8

J. Ka?ur, A. Handlovi?ová, and M. Ka?urová, Solution of Nonlinear Diffusion Problems by Linear Approximation Schemes, SIAM Journal on Numerical Analysis, vol.30, issue.6, pp.1703-1722, 1993.
DOI : 10.1137/0730087

I. Karatzas and S. E. Shreve, Brownian motion and stochastic calculus, Graduate Texts in Mathematics, vol.113, 1991.

H. P. Jr and . Mckean, Propagation of chaos for a class of non-linear parabolic equations., Stochastic Differential Equations (Lecture Series in Differential Equations, Session 7, Catholic Univ, Air Force Office Sci. Res, pp.41-57, 1967.

S. Méléard and S. Roelly-coppoletta, A propagation of chaos result for a system of particles with moderate interaction, Stochastic Process, Appl, vol.26, pp.317-332, 1987.

K. Oelschläger, A law of large numbers for moderately interacting diffusion processes A fluctuation theorem for moderately interacting diffusion processes Simulation of the solution of a viscous porous medium equation by a particle method, Z. Wahrsch. Verw. Gebiete Probab. Theory Related Fields SIAM J. Numer. Anal, vol.69, issue.35, pp.279-322, 1985.

L. Pareschi and G. Russo, Implicit-Explicit Runge-Kutta schemes and applications to hyperbolic systems with relaxation, J. Sci. Comput, vol.25, pp.129-155, 2005.

E. Parzen, On Estimation of a Probability Density Function and Mode, The Annals of Mathematical Statistics, vol.33, issue.3, pp.1065-1076, 1962.
DOI : 10.1214/aoms/1177704472

R. Philipowski, Interacting diffusions approximating the porous medium equation and propagation of chaos, Stochastic Process, Appl, vol.117, pp.526-538, 2007.

I. S. Pop and W. Yong, A numerical approach to degenerate parabolic equations, Numerische Mathematik, vol.92, issue.2, pp.357-381, 2002.
DOI : 10.1007/s002110100330

M. Rudemo, Empirical choice of histograms and kernel density estimators, Scand, J. Statist, vol.9, pp.65-78, 1982.

D. W. Scott and G. R. Terrell, Biased and Unbiased Cross-Validation in Density Estimation, Journal of the American Statistical Association, vol.9, issue.400, pp.1131-1146, 1987.
DOI : 10.1214/aoms/1177696810

S. J. Sheather and M. C. Jones, A reliable data-based bandwidth selection method for kernel density estimation, J. Roy. Statist. Soc. Ser. B, vol.53, pp.683-690, 1991.

R. E. Showalter, Monotone operators in Banach space and nonlinear partial differential equations, Mathematical Surveys and Monographs 49, 1997.

C. Shu, Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, Advanced numerical approximation of nonlinear hyperbolic equations (Cetraro, Lecture Notes in Math. 1697, pp.325-432, 1997.

B. W. Silverman, Density estimation for statistics and data analysis, Monographs on Statistics and Applied Probability, 1986.

D. W. Stroock and S. R. Varadhan, Multidimensional diffusion processes, Classics in Mathematics, 2006.
DOI : 10.1007/3-540-28999-2

A. S. Sznitman, Topics in propagation of chaos, Lecture Notes in Math, vol.22, issue.1, pp.165-251, 1991.
DOI : 10.1070/SM1974v022n01ABEH001689

G. R. Terrell, The Maximal Smoothing Principle in Density Estimation, Journal of the American Statistical Association, vol.9, issue.410, pp.470-477, 1990.
DOI : 10.1080/01621459.1985.10477163

G. R. Terrell and D. W. Scott, Oversmoothed Nonparametric Density Estimates, Journal of the American Statistical Association, vol.21, issue.389, pp.209-214, 1985.
DOI : 10.1214/aos/1176345341

M. P. Wand and M. C. Jones, Kernel smoothing, Monographs on Statistics and Applied Probability 60, 1995.

M. Woodroofe, On Choosing a Delta-Sequence, The Annals of Mathematical Statistics, vol.41, issue.5, pp.1665-1671, 1970.
DOI : 10.1214/aoms/1177696810

J. F. Grotowski, Z. I. Botev, and D. P. Kroese, Kernel density estimation via diffusion, Submitted to the Annals of statistics, 2007.

N. Belaribi and . Laboratoire-d-'analyse, Université Paris 13, 99, avenue Jean-Baptiste Clément, F-93430 Villetaneuse and ENSTA ParisTech, Géométrie et Applications (LAGA), vol.32

F. Russo and E. Paristech, Boulevard Victor , F-75739 Paris Cedex 15, INRIA Rocquencourt and Cermics Ecole des Ponts et Chaussées, Projet MATHFI Domaine de Voluceau, BP 105 F-78153 Le Chesnay Cedex