L. Bertini, A. De-sole, D. Gabrielli, G. Jona-lasinio, and C. Landim, Macroscopic Fluctiation Theory, vol.87, p.593, 2015.

C. Bernardin and S. Olla, Fourier law for a microscopic model of heat conduction, Journ. Stat. Phys, vol.121, pp.271-289, 2005.

P. Billingsley, Convergence of Probability measures Wiley, 1968.

A. De-masi, S. Olla, and E. Presutti, A note on Fick's law with phase transitions, Journal Statistical Physics, vol.175, pp.203-224, 2019.

B. Derrida, J. L. Lebowitz, and E. R. Speer, Free Energy Functional for Nonequilibrium Systems: An Exactly Solvable Case, Phys. Rev. Lett, vol.87, p.150601, 2001.

B. Derrida, J. L. Lebowitz, and E. R. Speer, Large Deviation of the Density Profile in the Steady State of the Open Symmetric Simple Exclusion Process Journ, Stat. Phys, vol.107, p.599634, 2002.

G. Gallavotti, The phase separation line in the two-dimensional Ising model, Commun, Math. Phys, vol.27, pp.103-136, 1972.

C. Kipnis, C. Marchioro, and E. Presutti, Heat flow in an exactly solvable model, Journ. Stat. Phys, vol.27, p.6574, 1982.

D. Ioffe, S. Shlosman, and Y. Velenik, An invariance Principle to Ferrari-Spohn diffusions, Commun. Math. Phys, vol.336, p.13, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01255370

H. Hanche-olsen and H. Holden, The KolmogorovRiesz compactness theorem, Expositiones Mathematicae, vol.28, pp.385-394, 2010.

M. G. Krein and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space, Ann. Math.Soc. Transl, vol.26, p.128, 1950.

G. F. Lawler and V. Limic, Random walk: a modern introduction Cambridge Studies in Advanced Mathematics 123, 2010.

A. De-masi, Dipartimento di Ingegneria e Scienze dell'Informazione e Matematica

S. Olla, :. Cnrs, C. , and U. Paris-dauphine, , vol.75016