I. Babu?ka, F. Nobile, and R. Tempone, A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data, SIAM Journal on Numerical Analysis, vol.45, issue.3, pp.1005-1034, 2007.
DOI : 10.1137/050645142

C. Bardos, ??quation de transport. Th??orie spectrale et approximation de la diffusion, Journ??es ??quations aux d??riv??es partielles, pp.1-10, 1982.
DOI : 10.5802/jedp.248

A. Bourgeat, M. Kern, S. Schumacher, and J. Talandier, The COUPLEX Test Cases: Nuclear Waste Disposal Simulation, Computational Geosciences, vol.8, issue.2, pp.83-89, 2004.
DOI : 10.1023/B:COMG.0000035073.03009.5d
URL : https://hal.archives-ouvertes.fr/hal-01315461

M. Bossy, N. Champagnat, S. Maire, and D. Talay, Probabilistic interpretation and random walk on spheres algorithms for the Poisson-Boltzmann equation in molecular dynamics, ESAIM: Mathematical Modelling and Numerical Analysis, vol.44, issue.5, 2010.
DOI : 10.1051/m2an/2010050
URL : https://hal.archives-ouvertes.fr/inria-00459411

R. S. Cantrell and C. Cosner, Diffusion Models for Population Dynamics Incorporating Individual Behavior at Boundaries: Applications to Refuge Design, Theoretical Population Biology, vol.55, issue.2, pp.189-207, 1999.
DOI : 10.1006/tpbi.1998.1397

E. Chénier, R. Eymard, and M. Kern, A Finite Volume Scheme for the Transport of Radionucleides in Porous Media, Computational Geosciences, vol.8, issue.2, pp.163-172, 2004.
DOI : 10.1023/B:COMG.0000035077.63408.71

G. Dagan, Flow and Transport in Porous Formations, 1989.
DOI : 10.1007/978-3-642-75015-1

R. Dautray and J. Lions, Évolution : numérique, transport, of Analyse Mathématique et Calcul Numérique pour les Sciences et Techniques. Masson, 1987.

M. Deaconu and A. Lejay, A Random Walk on Rectangles Algorithm, Methodology and Computing in Applied Probability, vol.24, issue.2, pp.135-151, 2006.
DOI : 10.1007/s11009-006-7292-3
URL : https://hal.archives-ouvertes.fr/inria-00092424

M. Deaconu and A. Lejay, Simulation of diffusions by means of importance sampling paradigm, The Annals of Applied Probability, vol.20, issue.4, 2010.
DOI : 10.1214/09-AAP659
URL : https://hal.archives-ouvertes.fr/inria-00126339

M. Decamps, A. De-schepper, M. Goovaerts, and W. Schoutens, A note on some new perpetuities, Scandinavian Actuarial Journal, vol.52, issue.4, pp.261-270, 2005.
DOI : 10.1137/1135018

D. Estep, S. Målqvist, and . Tavener, Nonparametric Density Estimation for Randomly Perturbed Elliptic Problems I: Computational Methods, A Posteriori Analysis, and Adaptive Error Control, SIAM Journal on Scientific Computing, vol.31, issue.4, pp.2935-2995, 2009.
DOI : 10.1137/080731670

D. Estep, S. Målqvist, and . Tavener, Nonparametric density estimation for randomly perturbed elliptic problems II: Applications and adaptive modeling, International Journal for Numerical Methods in Engineering, vol.41, issue.6??????7, pp.846-867, 2009.
DOI : JCOMP-D-08-00261

P. Étoré, Approximation de processus de diffusion à coefficients discontinus en dimension un et applications à la simulation, 2006.

P. Étoré, On random walk simulation of one-dimensional diffusion processes with discontinuous coefficients, Electronic Journal of Probability, vol.11, issue.0, pp.249-275, 2006.
DOI : 10.1214/EJP.v11-311

P. Étoré and A. Lejay, A Donsker theorem to simulate one-dimensional processes with measurable coefficients, ESAIM: Probability and Statistics, vol.11, pp.301-326, 2007.
DOI : 10.1051/ps:2007021

O. Faugeras, F. Clément, R. Deriche, R. Keriven, T. Papadopoulo et al., The inverse EEG and MEG problems: The adjoint state approach I: The continuous case, 1999.
URL : https://hal.archives-ouvertes.fr/inria-00077112

O. Faure, Simulation du mouvement brownien et des diffusions, 1992.
URL : https://hal.archives-ouvertes.fr/tel-00523258

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, 1998.

T. Hillen and G. Othmer, The diffusion limit of transport equations derived from velocity-jump processes, SIAM J. Appl. Math, vol.61, issue.3, pp.751-775, 2000.

C. Hwang, M. Mascagni, and J. A. Given, A Feynman???Kac path-integral implementation for Poisson???s equation using an h-conditioned Green???s function, 3rd IMACS Seminar on Monte Carlo Methods?MCM, pp.3-6347, 2001.
DOI : 10.1016/S0378-4754(02)00224-0

P. E. Kloeden and E. Platen, Numerical solution of stochastic differential equations, 1992.

O. A. Lady?enskaja, V. Ja, and N. N. Rivkind, Ural'ceva. Solvability of diffraction problems in the classical sense, Trudy Mat. Inst. Steklov, vol.92, pp.116-146, 1966.

J. Gall, One ??? dimensional stochastic differential equations involving the local times of the unknown process, Stochastic Analysis and Applications, pp.51-82, 1985.
DOI : 10.1512/iumj.1975.24.24047

A. Lejay, Méthodes probabilistes pour l'homogénéisation des opérateurs sous forme-divergence : cas linéaires et semi-linéaires, 2000.

A. Lejay and S. Maire, Computing the principal eigenvalue of the Laplace operator by a stochastic method, Mathematics and Computers in Simulation, vol.73, issue.6, pp.351-363, 2006.
DOI : 10.1016/j.matcom.2006.06.011
URL : https://hal.archives-ouvertes.fr/inria-00092408

A. Lejay and S. Maire, Computing the principal eigenelements of some linear operators using a branching Monte Carlo method, Journal of Computational Physics, vol.227, issue.23, pp.9794-9806, 2008.
DOI : 10.1016/j.jcp.2008.07.018
URL : https://hal.archives-ouvertes.fr/hal-01479830

A. Lejay and M. Martinez, A scheme for simulating one-dimensional diffusion processes with discontinuous coefficients, The Annals of Applied Probability, vol.16, issue.1, pp.107-139, 2006.
DOI : 10.1214/105051605000000656
URL : https://hal.archives-ouvertes.fr/inria-00000410

N. Limi?, Markov Jump Processes Approximating a Nonsymmetric Generalized Diffusion, 2008.

S. Maire, Réduction de variance pour l'intégration numérique et pour le calcul critique en transport neutronique, 2001.

S. Maire and D. Talay, On a Monte Carlo method for neutron transport criticality computations, IMA Journal of Numerical Analysis, vol.26, issue.4, pp.657-685, 2006.
DOI : 10.1093/imanum/drl008
URL : https://hal.archives-ouvertes.fr/hal-01479840

M. Martinez, Interprétations probabilistes d'opérateurs sous forme divergence et analyse de méthodes numériques associées, 2004.

M. Martinez and D. Talay, Discr??tisation d'??quations diff??rentielles stochastiques unidimensionnelles ?? g??n??rateur sous forme divergence avec coefficient discontinu, Comptes Rendus Mathematique, vol.342, issue.1, pp.51-56, 2006.
DOI : 10.1016/j.crma.2005.10.025

M. Mascagni and N. A. Simonov, Monte Carlo Methods for Calculating Some Physical Properties of Large Molecules, SIAM Journal on Scientific Computing, vol.26, issue.1, pp.339-357, 2004.
DOI : 10.1137/S1064827503422221

G. N. Milstein and M. V. Tretyakov, Simulation of a space-time bounded diffusion, The Annals of Applied Probability, vol.9, issue.3, pp.732-779, 1999.
DOI : 10.1214/aoap/1029962812

M. E. Muller, Some Continuous Monte Carlo Methods for the Dirichlet Problem, The Annals of Mathematical Statistics, vol.27, issue.3, pp.569-589, 1956.
DOI : 10.1214/aoms/1177728169

O. Ovaskainen and S. J. Cornell, Biased movement at a boundary and conditional occupancy times for diffusion processes, Journal of Applied Probability, vol.52, issue.53, pp.557-580, 2003.
DOI : 10.1006/tpbi.1998.1397

J. M. Ramirez, E. A. Thomann, E. C. Waymire, R. Haggerty, and W. B. , A Generalized Taylor???Aris Formula and Skew Diffusion, Multiscale Modeling & Simulation, vol.5, issue.3, pp.786-801, 2006.
DOI : 10.1137/050642770

P. and S. Tonou, Méthodes Numériques Probabilistes pour la Résolution d'Équations du Transport et pour l'Évaluation d'Options Exotiques, 1997.

D. W. Stroock, Diffusion semigroups corresponding to uniformly elliptic divergence form operators, Séminaire de Probabilités XXII, pp.316-347, 1988.
DOI : 10.2307/2372841

M. Zhang, Calculation of Diffusive Shock Acceleration of Charged Particles by Skew Brownian Motion, The Astrophysical Journal, vol.541, issue.1, pp.428-435, 2000.
DOI : 10.1086/309429

T. Projet, I. Élie-cartan-de-nancynancy-université, and C. , Campus scientifique, BP 239, 54506 Vandoeuvre-lès- Nancy CEDEX, France E-mail address: Antoine.Lejay@iecn.u-nancy.fr ISITV