Abstract
Prolongating our previous paper on the Einstein relation, we study the motion of a particle diffusing in a random reversible environment when subject to a small external forcing. In order to describe the long time behavior of the particle, we introduce the notions of steady state and weak steady state. We establish the continuity of weak steady states for an ergodic and uniformly elliptic environment. When the environment has finite range of dependence, we prove the existence of the steady state and weak steady state and compute its derivative at a vanishing force. Thus we obtain a complete ‘fluctuation–dissipation Theorem’ in this context as well as the continuity of the effective variance.
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Aronson, D.G.: Non-negative solutions of linear parabolic equations. Ann. Scuola Norm. Sup. di Pisa 22(4), 607–694 (1968)
Baladi, V.: Linear response, or else. ICM Seoul, 2014, Proceedings, Vol III, 525–545 2014
Campos, D., Ramírez, A.F.: Asymptotic expansion of the invariant measure for ballistic random walks in random environment in the low disorder regime. Ann. Probab. 45, 4675–4699 (2015)
De Masi, A., Ferrari, P., Goldstein, S., Wick, W.D.: An invariance principle for reversible Markov processes. Applications to random motions in random environments. J. Stat. Phys. 55(3/4), 787–855 (1989)
Einstein, A.: Investigations on the theory of the Brownian movement. Edited with notes by R. Fürth. Translated by A. D. Cowper. Dover Publications, Inc., New York 1956
Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet forms and symmetric Markov processes. de Gruyter Studies in Mathematics, 19. Walter de Gruyter, Berlin 1994
Gantert, N., Mathieu, P., Piatnitski, A.: Einstein relation for reversible diffusions in a random environment. Comm. Pure Appl. Math. 65(2), 187–228 (2012)
Gantert, N., Guo, X., Nagel, J.: Einstein relation and steady states for the random conductance model. Ann. Probab., to appear 2015
Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer-Verlag, Berlin (2001)
Gloria, A., Marahrens, D.: Annealed estimates on the Green functions and uncertainty quantification Ann. Inst. H. Poincaré Anal. Non Linéaire, to appear
Gloria, A., Otto, F.: Quantitative results on the corrector equation in stochastic homogenization. J. Eur. Math. Soc., to appear
Hairer, M., Majda, A.J.: A simple framework to justify linear response theory. Nonlinearity. 23(4), 909–922 (2010)
Jikov, V.V., Kozlov, S.M., Oleinik, O.A.: Homogenization of differential operators and integral functionals. Springer-Verlag, Berlin (1994)
Kipnis, C., Varadhan, S.R.S.: A central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions. Comm. Math. Phys. 104, 1–19 (1986)
Kesten, H.: A renewal theorem for random walk in a random environment. Symposia Pure. Math. 31, 67–77 (1977)
Komorowski, T., Krupa, G.: The existence of a steady state for a perturbed symmetric random walk on a random lattice. Probab. Math. Statist. 24(1), Acta Univ. Wratislav. No. 2646, 121144 2004
Komorowski, T., Krupa, G.: The existence of a steady state for a diffusion in a random environment with an external forcing. Multi scale problems and asymptotic analysis, 181194, GAKUTO Internat. Ser. Math. Sci. Appl., 24, Gakktosho, Tokyo, 2006 2006
Komorowski, T., Landim, C., Olla, S.: Fluctuations in Markov Processes. Springer-Verlag, Time Symmetry and Martingale Approximation (2012)
Komorowski, T., Olla, S.: On Mobility and Einstein Relation for Tracers in Time-Mixing Random Environments. J. Stat Phys. 118, 407–435 (2005)
Kozlov, S.M.: Averaging of Random Operators. Sb. Math. 37(2), 167–180 (1980)
Kozlov, S.M.: The method of averaging and walks in inhomogeneous environments Russian Math. Surveys 40(2), 73–145 (1985)
Kubo, R., Toda, M., Hashitsume, N.: Statistical Physics. II. Nonequilibrium Statistical Mechanics. Springer Series in Solid-State Sciences, 31. Springer-Verlag, Berlin 1985
Lebowitz, J.L., Rost, H.: The Einstein relation for the displacement of a test particle in a random environment. Stochastic Process. Appl. 54(2), 183–196 (1994)
Mourrat, J.-C.: Variance decay for functionals of the environment viewed by the particle. Ann. I. H. Poincaré-PR 47(1), 294–327 (2001)
Osada, H.: Homogenization of diffusion processes with randomstationary coefficients. Probability theory and mathematical statistics (Tbilisi, 1982), 507–517, Lecture Notes in Math., 1021, Springer, Berlin, 1983 1982
Papanicolaou, G., Varadhan, S.R.S.: Diffusion with Random Coefficients. Essays in Honor of C.R. Rao, ed. by G. Kallianpur, P.R. Krishnajah, J.K. Gosh pp. 547–552. Amsterdam: North Holland 1982 1982
Perrin, J.: Les atomes. First edited in 1913 by Félix Alcan. Reissued in 2014 by CNRS Editions 1913
Revuz, D., Yor, M.: Continuous martingales and Brownian motion. Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences) 293. Springer, Berlin 1994
Shen, L.: On ballistic diffusions in random environment Ann. I. H. Poincaré-PR 39(5), 839–876 (2003)
Sznitman, A., Zerner, M.: A law of large numbers for random walks in random environment. Ann. Probab. 27, 1851–1869 (1999)
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Mathieu, P., Piatnitski, A. Steady States, Fluctuation–Dissipation Theorems and Homogenization for Reversible Diffusions in a Random Environment. Arch Rational Mech Anal 230, 277–320 (2018). https://doi.org/10.1007/s00205-018-1245-1
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DOI: https://doi.org/10.1007/s00205-018-1245-1