Symmetric simple exclusion process in dynamic environment: hydrodynamics. (English) Zbl 1469.60345
Summary: We consider the symmetric simple exclusion process in \(\mathbb{Z}^d\) with quenched bounded dynamic random conductances and prove its hydrodynamic limit in path space. The main tool is the connection, due to the self-duality of the process, between the invariance principle for single particles starting from all points and the macroscopic behavior of the density field. While the hydrodynamic limit at fixed macroscopic times is obtained via a generalization to the time-inhomogeneous context of the strategy introduced in [K. Nagy, Period. Math. Hung. 45, No. 1–2, 101–120 (2002; Zbl 1064.60202)], in order to prove tightness for the sequence of empirical density fields we develop a new criterion based on the notion of uniform conditional stochastic continuity, following [S. R. S. Varadhan, Stochastic processes. Providence, RI: American Mathematical Society (AMS); New York, NY: Courant Institute of Mathematical Sciences (2007; Zbl 1133.60004)]. In conclusion, we show that uniform elliptic dynamic conductances provide an example of environments in which the so-called arbitrary starting point invariance principle may be derived from the invariance principle of a single particle starting from the origin. Therefore, our hydrodynamics result applies to the examples of quenched environments considered in, e.g., [S. Andres, Ann. Inst. Henri Poincaré, Probab. Stat. 50, No. 2, 352–374 (2014; Zbl 1290.60109); S. Andres et al., Ann. Probab. 46, No. 1, 302–336 (2018; Zbl 1429.60076); Electron. J. Probab. 24, Paper No. 87, 29 p. (2019; Zbl 1466.60216)] in combination with the hypothesis of uniform ellipticity.
MSC:
| 60K35 | Interacting random processes; statistical mechanics type models; percolation theory |
| 60K37 | Processes in random environments |
| 60J28 | Applications of continuous-time Markov processes on discrete state spaces |
| 60F17 | Functional limit theorems; invariance principles |
| 82C22 | Interacting particle systems in time-dependent statistical mechanics |
Keywords:
hydrodynamic limit; symmetric simple exclusion process; dynamic random conductances; arbitrary starting point invariance principle; tightness criterionReferences:
| [1] | S. Andres, Invariance principle for the random conductance model with dynamic bounded conductances, Ann. Inst. Henri Poincaré Probab. Stat. 50 (2014), no. 2, 352-374. · Zbl 1290.60109 · doi:10.1214/12-AIHP527 |
| [2] | S. Andres, M. T. Barlow, J.-D. Deuschel, and B. M. Hambly, Invariance principle for the random conductance model, Probab. Theory Related Fields 156 (2013), no. 3-4, 535-580. · Zbl 1356.60174 |
| [3] | S. Andres, A. Chiarini, J.-D. Deuschel, and M. Slowik, Quenched invariance principle for random walks with time-dependent ergodic degenerate weights, Ann. Probab. 46 (2018), no. 1, 302-336. · Zbl 1429.60076 · doi:10.1214/17-AOP1186 |
| [4] | L. Avena, O. Blondel, and A. Faggionato, Analysis of random walks in dynamic random environments via \(L^2\)-perturbations, Stochastic Process. Appl. 128 (2018), no. 10, 3490-3530. · Zbl 1409.60147 · doi:10.1016/j.spa.2017.11.010 |
| [5] | P. Billingsley, Convergence of probability measures. Second edition. Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, Inc., New York, 1999, A Wiley-Interscience Publication. · Zbl 0944.60003 |
| [6] | M. Biskup, An invariance principle for one-dimensional random walks among dynamical random conductances, Electron. J. Probab. 24 (2019), Paper No. 87, 29. · Zbl 1466.60216 · doi:10.1214/19-EJP348 |
| [7] | M. Biskup and P.-F. Rodriguez, Limit theory for random walks in degenerate time-dependent random environments, J. Funct. Anal. 274 (2018), no. 4, 985-1046. · Zbl 1405.60150 · doi:10.1016/j.jfa.2017.12.002 |
| [8] | B. Böttcher, Feller evolution systems: generators and approximation, Stoch. Dyn. 14 (2014), no. 3, 1350025, 15. · Zbl 1321.47095 |
| [9] | G. Carinci, C. Giardinà, C. Giberti, and F. Redig, Duality for stochastic models of transport, J. Stat. Phys. 152 (2013), no. 4, 657-697. · Zbl 1345.60083 · doi:10.1007/s10955-013-0786-9 |
| [10] | Z.-Q. Chen, D. A. Croydon, and T. Kumagai, Quenched invariance principles for random walks and elliptic diffusions in random media with boundary, Ann. Probab. 43 (2015), no. 4, 1594-1642. · Zbl 1338.60098 · doi:10.1214/14-AOP914 |
| [11] | A. De Masi and E. Presutti, Mathematical methods for hydrodynamic limits, Lecture Notes in Mathematics, vol. 1501, Springer-Verlag, Berlin, 1991. · Zbl 0754.60122 |
| [12] | T. Delmotte and J.-D. Deuschel, On estimating the derivatives of symmetric diffusions in stationary random environment, with applications to \(\nabla \phi\) interface model, Probab. Theory Related Fields 133 (2005), no. 3, 358-390. · Zbl 1083.60082 · doi:10.1007/s00440-005-0430-y |
| [13] | J.-D. Deuschel and M. Slowik, Invariance principle for the one-dimensional dynamic random conductance model under moment conditions, Stochastic analysis on large scale interacting systems, RIMS Kôkyûroku Bessatsu, B59, Res. Inst. Math. Sci. (RIMS), Kyoto, 2016, pp. 69-84. · Zbl 1367.60027 |
| [14] | K.-J. Engel and R. Nagel, One-parameter semigroups for linear evolution equations, Graduate Texts in Mathematics, vol. 194, Springer-Verlag, New York, 2000, With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt. · Zbl 0952.47036 |
| [15] | S. N. Ethier and T. G. Kurtz, Markov processes: Characterization and convergence, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1986. · Zbl 0592.60049 |
| [16] | L. C. Evans, Partial differential equations, second ed., Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 2010. · Zbl 1194.35001 |
| [17] | A. Faggionato, Bulk diffusion of 1D exclusion process with bond disorder, Markov Process. Related Fields 13 (2007), no. 3, 519-542. · Zbl 1144.60058 |
| [18] | A. Faggionato, Random walks and exclusion processes among random conductances on random infinite clusters: homogenization and hydrodynamic limit, Electron. J. Probab. 13 (2008), no. 73, 2217-2247. · Zbl 1189.60172 · doi:10.1214/EJP.v13-591 |
| [19] | A. Faggionato, Hydrodynamic limit of symmetric exclusion processes in inhomogeneous media, arXiv:1003.5521 (2010). |
| [20] | A. Faggionato, Hydrodynamic limit of zero range processes among random conductances on the supercritical percolation cluster, Electron. J. Probab. 15 (2010), no. 10, 259-291. · Zbl 1201.60093 · doi:10.1214/EJP.v15-748 |
| [21] | A. Faggionato, M. Jara, and C. Landim, Hydrodynamic behavior of 1D subdiffusive exclusion processes with random conductances, Probab. Theory Related Fields 144 (2009), no. 3-4, 633-667. · Zbl 1169.60326 · doi:10.1007/s00440-008-0157-7 |
| [22] | A. Faggionato and F. Martinelli, Hydrodynamic limit of a disordered lattice gas, Probab. Theory Related Fields 127 (2003), no. 4, 535-608. · Zbl 1052.60083 |
| [23] | S. Floreani, F. Redig, and F. Sau, Hydrodynamics for the partial exclusion process in random environment, arXiv:1911.12564 (2019). |
| [24] | J. Fritz, Hydrodynamics in a symmetric random medium, Comm. Math. Phys. 125 (1989), no. 1, 13-25. · Zbl 0682.76001 · doi:10.1007/BF01217766 |
| [25] | G. Giacomin, J. L. Lebowitz, and E. Presutti, Deterministic and stochastic hydrodynamic equations arising from simple microscopic model systems, Stochastic partial differential equations: six perspectives, Math. Surveys Monogr., vol. 64, Amer. Math. Soc., Providence, RI, 1999, pp. 107-152. · Zbl 0927.60060 |
| [26] | G. Giacomin, S. Olla, and H. Spohn, Equilibrium fluctuations for \(\nabla \phi\) interface model, Ann. Probab. 29 (2001), no. 3, 1138-1172. · Zbl 1017.60100 · doi:10.1214/aop/1015345767 |
| [27] | P. Gonçalves and M. Jara, Scaling limits for gradient systems in random environment, J. Stat. Phys. 131 (2008), no. 4, 691-716. · Zbl 1144.82043 · doi:10.1007/s10955-008-9509-z |
| [28] | G. Grimmett, Percolation. Second edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 321, Springer-Verlag, Berlin, 1999. · Zbl 0926.60004 |
| [29] | T. E. Harris, Nearest-neighbor Markov interaction processes on multidimensional lattices, Advances in Math. 9 (1972), 66-89. · Zbl 0267.60107 · doi:10.1016/0001-8708(72)90030-8 |
| [30] | K. Itô, Continuous additive \({\mathscr{S}}^{\prime } \)-processes, Stochastic differential systems (Proc. IFIP-WG 7/1 Working Conf., Vilnius, 1978), Lecture Notes in Control and Information Sci., vol. 25, Springer, Berlin-New York, 1980, pp. 143-151. |
| [31] | M. Jara and C. Landim, Quenched non-equilibrium central limit theorem for a tagged particle in the exclusion process with bond disorder, Ann. Inst. Henri Poincaré Probab. Stat. 44 (2008), no. 2, 341-361. · Zbl 1195.60124 · doi:10.1214/07-AIHP112 |
| [32] | A. Joffe and M. Métivier, Weak convergence of sequences of semimartingales with applications to multitype branching processes, Adv. in Appl. Probab. 18 (1986), no. 1, 20-65. · Zbl 0595.60008 |
| [33] | O. Kallenberg, Foundations of modern probability. Second edition. Probability and its Applications (New York), Springer-Verlag, New York, 2002. · Zbl 0996.60001 |
| [34] | C. Kipnis and C. Landim, Scaling limits of interacting particle systems, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 320, Springer-Verlag, Berlin, 1999. · Zbl 0927.60002 |
| [35] | T. G. Kurtz, Extensions of Trotter’s operator semigroup approximation theorems, J. Functional Analysis 3 (1969), 354-375. · Zbl 0174.18401 · doi:10.1016/0022-1236(69)90031-7 |
| [36] | J. L. Lebowitz and H. Spohn, Microscopic basis for Fick’s law for self-diffusion, J. Statist. Phys. 28 (1982), no. 3, 539-556. · Zbl 0512.60075 |
| [37] | P. A. W. Lewis and G. S. Shedler, Simulation of nonhomogeneous Poisson processes by thinning, Naval Res. Logist. Quart. 26 (1979), no. 3, 403-413. · Zbl 0497.60003 |
| [38] | T. M. Liggett, Interacting particle systems, Classics in Mathematics, Springer-Verlag, Berlin, 2005, Reprint of the 1985 original. · Zbl 1103.82016 |
| [39] | I. Mitoma, Tightness of probabilities on \({\mathcal{C}}([0,1];{ \mathscr{S}}^{\prime } )\) and \({\mathcal{D}}([0,1];{\mathscr{S}}^{ \prime } )\), Ann. Probab. 11 (1983), no. 4, 989-999. · Zbl 0527.60004 · doi:10.1214/aop/1176993447 |
| [40] | J.-C. Mourrat and F. Otto, Anchored Nash inequalities and heat kernel bounds for static and dynamic degenerate environments, J. Funct. Anal. 270 (2016), no. 1, 201-228. · Zbl 1330.35010 · doi:10.1016/j.jfa.2015.09.020 |
| [41] | K. Nagy, Symmetric random walk in random environment in one dimension, Period. Math. Hungar. 45 (2002), no. 1-2, 101-120. · Zbl 1064.60202 · doi:10.1023/A:1022354131403 |
| [42] | G. C. Papanicolaou and S. R. S. Varadhan, Boundary value problems with rapidly oscillating random coefficients, Random fields, Vol. I, II (Esztergom, 1979), Colloq. Math. Soc. János Bolyai, vol. 27, North-Holland, Amsterdam-New York, 1981, pp. 835-873. · Zbl 0499.60059 |
| [43] | S. Peszat and J. Zabczyk, Stochastic partial differential equations with Lévy noise, An evolution equation approach. Encyclopedia of Mathematics and its Applications, vol. 113, Cambridge University Press, Cambridge, 2007. · Zbl 1205.60122 |
| [44] | J. Quastel, Diffusion of color in the simple exclusion process, Comm. Pure Appl. Math. 45 (1992), no. 6, 623-679. · Zbl 0769.60097 |
| [45] | F. Redig and F. Völlering, Random walks in dynamic random environments: a transference principle, Ann. Probab. 41 (2013), no. 5, 3157-3180. · Zbl 1277.82051 · doi:10.1214/12-AOP819 |
| [46] | D. Revuz and M. Yor, Continuous martingales and Brownian motion, third ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 293, Springer-Verlag, Berlin, 1999. · Zbl 0917.60006 |
| [47] | F. Spitzer, Interaction of Markov processes, Advances in Math. 5 (1970), 246-290 (1970). · Zbl 0312.60060 · doi:10.1016/0001-8708(70)90034-4 |
| [48] | D. W. Stroock and W. Zheng, Markov chain approximations to symmetric diffusions, Ann. Inst. H. Poincaré Probab. Statist. 33 (1997), no. 5, 619-649. · Zbl 0885.60065 |
| [49] | S. R. S. Varadhan, Nonlinear diffusion limit for a system with nearest neighbor interactions. II, Asymptotic problems in probability theory: stochastic models and diffusions on fractals (Sanda/Kyoto, 1990), Pitman Res. Notes Math. Ser., vol. 283, Longman Sci. Tech., Harlow, 1993, pp. 75-128. · Zbl 0793.60105 |
| [50] | S. R. S. Varadhan, Stochastic processes, Courant Lecture Notes in Mathematics, vol. 16, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2007. · Zbl 1133.60004 |
| [51] | A. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
