D. Applebaum, Lévy processes and stochastic calculus, Cambridge Studies in Advanced Mathematics, vol.116, 2009.

M. Arisawa, Homogenization of a Class of Integro-Differential Equations with L??vy Operators, Communications in Partial Differential Equations, vol.8, issue.7, pp.617-624, 2009.
DOI : 10.1080/03605309108820789

G. Barles, Asymptotic behavior of viscosity solutions of first Hamilton Jacobi equations, Ricerche Mat, vol.34, pp.227-260, 1985.
URL : https://hal.archives-ouvertes.fr/inria-00076153

G. Barles, E. Chasseigne, A. Ciomaga, and C. Imbert, Lipschitz regularity of solutions for mixed integro-differential equations, Journal of Differential Equations, vol.252, issue.11, pp.6012-6060, 2012.
DOI : 10.1016/j.jde.2012.02.013
URL : https://hal.archives-ouvertes.fr/hal-00608848

G. Barles, E. Chasseigne, and C. Imbert, Hölder continuity of solutions of second-order non-linear elliptic integro-differential equations, J. Eur. Math. Soc. (JEMS), vol.13, pp.1-26, 2011.

G. Barles and F. Da-lio, On the boundary ergodic problem for fully nonlinear equations in bounded domains with general nonlinear Neumann boundary conditions, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, vol.22, issue.5, pp.521-541, 2005.
DOI : 10.1016/j.anihpc.2004.09.001
URL : https://hal.archives-ouvertes.fr/hal-00002826

G. Barles, F. Da-lio, P. Lions, and P. E. Souganidis, Ergodic problems and periodic homogenization for fully nonlinear equations in half-space type domains with Neumann boundary conditions, Indiana University Mathematics Journal, vol.57, issue.5, pp.2355-2375, 2008.
DOI : 10.1512/iumj.2008.57.3363
URL : https://hal.archives-ouvertes.fr/hal-00426354

G. Barles and C. Imbert, Second-order elliptic integro-differential equations: viscosity solutions' theory revisited, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, vol.25, issue.3, pp.25-567, 2008.
DOI : 10.1016/j.anihpc.2007.02.007
URL : https://hal.archives-ouvertes.fr/hal-00130169

G. Barles, H. Ishii, and H. Mitake, On the Large Time Behavior of Solutions of Hamilton???Jacobi Equations Associated with Nonlinear Boundary Conditions, Archive for Rational Mechanics and Analysis, vol.237, issue.22, pp.515-558, 2012.
DOI : 10.1007/s00205-011-0484-1
URL : https://hal.archives-ouvertes.fr/hal-00623000

G. Barles and H. Mitake, A PDE Approach to Large-Time Asymptotics for Boundary-Value Problems for Nonconvex Hamilton???Jacobi Equations, Communications in Partial Differential Equations, vol.15, issue.1, pp.136-168, 2012.
DOI : 10.1016/j.physd.2008.05.009
URL : https://hal.archives-ouvertes.fr/hal-00542225

G. Barles, A. Porretta, and T. T. Tchamba, On the large time behavior of solutions of the Dirichlet problem for subquadratic viscous Hamilton???Jacobi equations, Journal de Math??matiques Pures et Appliqu??es, vol.94, issue.5, pp.94-497, 2010.
DOI : 10.1016/j.matpur.2010.03.006
URL : https://hal.archives-ouvertes.fr/hal-00441975

G. Barles and P. E. Souganidis, On the Large Time Behavior of Solutions of Hamilton--Jacobi Equations, SIAM Journal on Mathematical Analysis, vol.31, issue.4, pp.925-939, 2000.
DOI : 10.1137/S0036141099350869
URL : https://hal.archives-ouvertes.fr/hal-00623000

G. Barles and P. E. Souganidis, Some counterexamples on the asymptotic behavior of the solutions of Hamilton???Jacobi equations, Comptes Rendus de l'Acad??mie des Sciences - Series I - Mathematics, vol.330, issue.11, pp.963-968, 2000.
DOI : 10.1016/S0764-4442(00)00314-1

G. Barles and P. E. Souganidis, Space-Time Periodic Solutions and Long-Time Behavior of Solutions to Quasi-linear Parabolic Equations, SIAM Journal on Mathematical Analysis, vol.32, issue.6, pp.1311-1323, 2001.
DOI : 10.1137/S0036141000369344

A. Bensoussan, J. Lions, and G. Papanicolaou, Asymptotic analysis for periodic structures, of Studies in Mathematics and its Applications, 1978.

E. Chasseigne, M. Chaves, and J. D. Rossi, Asymptotic behavior for nonlocal diffusion equations, Journal de Math??matiques Pures et Appliqu??es, vol.86, issue.3, pp.86-271, 2006.
DOI : 10.1016/j.matpur.2006.04.005

A. Ciomaga, On the strong maximum principle for second order nonlinear parabolic integrodifferential equations Advances in Differential Equations, pp.635-671, 2012.

F. and D. Lio, Large time behavior of solutions to parabolic equations with Neumann boundary conditions, Journal of Mathematical Analysis and Applications, vol.339, issue.1, pp.384-398, 2008.
DOI : 10.1016/j.jmaa.2007.06.052

A. Davini and A. Siconolfi, A Generalized Dynamical Approach to the Large Time Behavior of Solutions of Hamilton--Jacobi Equations, SIAM Journal on Mathematical Analysis, vol.38, issue.2, pp.478-502, 2006.
DOI : 10.1137/050621955

N. Dirr and P. E. Souganidis, Large-Time Behavior for Viscous and Nonviscous Hamilton--Jacobi Equations Forced by Additive Noise, SIAM Journal on Mathematical Analysis, vol.37, issue.3, pp.777-796, 2005.
DOI : 10.1137/040611896

A. Fathi, Sur la convergence du semi-groupe de Lax-Oleinik, Comptes Rendus de l'Acad??mie des Sciences - Series I - Mathematics, vol.327, issue.3, pp.267-270, 1998.
DOI : 10.1016/S0764-4442(98)80144-4

N. Forcadel, C. Imbert, and R. Monneau, Homogenization of some particle systems with two-body interactions and of the dislocation dynamics, Discrete Contin, Dyn. Syst, vol.23, pp.785-826, 2009.

B. Franke, A functional non-central limit theorem for jump-diffusions with periodic coefficients driven by stable Lévy-noise Homogenization of random transport along periodic two-dimensional flows, Stochastic Process, J. Theoret. Probab. Appl, vol.2024, issue.119, pp.1087-1100, 2007.

T. Fujiwara and M. Tomisaki, Martingale approach to limit theorems for jump processes, Stochastics An International Journal of Probability and Stochastic Processes, vol.50, issue.1, pp.35-64, 1994.
DOI : 10.1080/17442509408833927

Y. Giga, Q. Liu, and H. Mitake, Large-time asymptotics for one-dimensional Dirichlet problems for Hamilton???Jacobi equations with noncoercive Hamiltonians, Journal of Differential Equations, vol.252, issue.2, pp.1263-1282, 2012.
DOI : 10.1016/j.jde.2011.10.010

M. Horie, T. Inuzuka, and H. Tanaka, Homogenization of certain one-dimensional discontinuous Markov processes, Hiroshima Math. J, vol.7, pp.629-641, 1977.

N. Ichihara and H. Ishii, Long-time Behavior of Solutions of Hamilton???Jacobi Equations with Convex and Coercive Hamiltonians, Archive for Rational Mechanics and Analysis, vol.80, issue.1, pp.383-419, 2009.
DOI : 10.1007/s00205-008-0170-0

C. Imbert, A non-local regularization of first order Hamilton???Jacobi equations, Journal of Differential Equations, vol.211, issue.1, pp.218-246, 2005.
DOI : 10.1016/j.jde.2004.06.001
URL : https://hal.archives-ouvertes.fr/hal-00176542

C. Imbert, R. Monneau, and E. Rouy, /??)-Periodic Hamiltonians Part II: Application to Dislocations Dynamics, Communications in Partial Differential Equations, vol.20, issue.3, pp.479-516, 2008.
DOI : 10.1209/epl/i2003-10175-2

H. Ishii, Asymptotic solutions for large time of Hamilton-Jacobi equations in Euclidean n space, Ann. Inst. H. Poincaré Anal Long-time asymptotic solutions of convex Hamilton-Jacobi equations with Neumann type boundary conditions, Non Linéaire Calc. Var. Partial Differential Equations, vol.2532, pp.231-266, 2008.

P. Lions, Generalized solutions of Hamilton-Jacobi equations, Research Notes in Mathematics, vol.69, 1982.

P. Lions, G. Papanicolaou, and S. Varadhan, Homogenization of Hamilton-Jacobi equations, 1987.
URL : https://hal.archives-ouvertes.fr/hal-00667310

H. Mitake, Asymptotic Solutions of Hamilton-Jacobi Equations with State Constraints, Applied Mathematics and Optimization, vol.94, issue.2, pp.393-410, 2008.
DOI : 10.1007/s00245-008-9041-1

G. Namah and J. Roquejoffre, Convergence to periodic fronts in a class of semilinear parabolic equations, NoDEA Nonlinear Differential Equations Appl, Remarks on the long time behaviour of the solutions of Hamilton-Jacobi equations, pp.521-536, 1997.

R. Rhodes and V. Vargas, Scaling limits for symmetric Itô-Lévy processes in random medium, Stochastic Process, Appl, vol.119, pp.4004-4033, 2009.
DOI : 10.1016/j.spa.2009.10.004
URL : http://doi.org/10.1016/j.spa.2009.10.004

J. Roquejoffre, Convergence to steady states or periodic solutions in a class of Hamilton???Jacobi equations, Journal de Math??matiques Pures et Appliqu??es, vol.80, issue.1, pp.80-85, 2001.
DOI : 10.1016/S0021-7824(00)01183-1

R. W. Schwab, Periodic Homogenization for Nonlinear Integro-Differential Equations, SIAM Journal on Mathematical Analysis, vol.42, issue.6, pp.2652-2680, 2010.
DOI : 10.1137/080737897

T. and T. Tchamba, Large time behavior of solutions of viscous Hamilton-Jacobi equations with superquadratic Hamiltonian, Asymptot. Anal, vol.66, pp.161-186, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00371068

M. Tomisaki, Homogenization of cadlag processes, Journal of the Mathematical Society of Japan, vol.44, issue.2, pp.281-305, 1992.
DOI : 10.2969/jmsj/04420281

M. Laboratoire-de, C. Et-physique-théorique, and . Umr, 37200 Tours, France E-mail address: barles@lmpt.univ-tours Fédération Denis Poisson Parc de Grandmont, 37200 Tours, France E-mail address: emmanuel.chasseigne@lmpt.univ-tours, fr Laboratoire de Mathématiques et Physique Théorique, CNRS UMR 7350 E-mail address: adina@math.uchicago.edu CNRS, Laboratoire d'Analyse et de Mathématiques Appliquées avenue du général de Gaulle 94010 Créteil cedex France E-mail address, p.61