# zbMATH — the first resource for mathematics

Exclusion processes in higher dimensions: stationary measures and convergence. (English) Zbl 1099.60067
The authors consider the exclusion process on $$Z^{d}$$ with $$d>1$$. Firstly, one gives a necessary and sufficient condition for a product measure to be stationary for the process. One applies then this condition to the case of a translation system on $$Z^{d}$$. This allows to construct many examples of stationary product measures that are neither homogeneous nor reversible. Examples for a random walk on a homogeneous tree and on a rooted tree are given. Then the authors study stationary measures which are invariant under translations in all directions orthogonal to a fixed nonzero $$v\in Z^{d}$$ and prove a number of convergence results for the measure of the exclusion process. Applications of hydrodynamical results are used to obtain explicit convergence results. Some open problems are also stated.

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory
hydrodynamics
Full Text:
##### References:
 [1] Bramson, M., Liggett, T. M. and Mountford, T. (2002). Characterization of stationary measures for one-dimensional exclusion processes. Ann. Probab. 30 1539–1575. · Zbl 1039.60086 · doi:10.1214/aop/1039548366 [2] Bramson, M. and Mountford, T. (2002). Stationary blocking measures for one-dimensional nonzero mean exclusion processes. Ann. Probab. 30 1082–1130. · Zbl 1042.60062 · doi:10.1214/aop/1029867122 [3] Choquet, G. and Deny, J. (1960). Sur l’equation de convolution $$\mu=\mu*\sigma$$. C. R. Acad. Sci. Paris 250 799–801. · Zbl 0093.12802 [4] Evans, L. (1998). Partial Differential Equations . Amer. Math. Soc., Providence, RI. · Zbl 0902.35002 [5] Ferrari, P. A., Lebowitz, J. L. and Speer, E. (2001). Blocking measures for asymmetric exclusion processes via coupling. Bernoulli 7 935–950. · Zbl 1002.60100 · doi:10.2307/3318627 [6] Georgii, H. O. (1979). Canonical Gibbs Measures . Lecture Notes in Math. 760 . Springer, Berlin. · Zbl 0409.60094 · doi:10.1007/BFb0068557 [7] Jung, P. (2003). Extremal reversible measures for the exclusion process. J. Statist. Phys. 112 165–191. · Zbl 1025.60045 · doi:10.1023/A:1023679620839 [8] Kipnis, C. and Landim, C. (1999). Scaling Limits of Interacting Particle Systems . Springer, Berlin. · Zbl 0927.60002 [9] Kruzkov, S. N. (1970). First order quasilinear equations in several independent variables. Mat. Sb. 123 228–255. (English translation in Math. USSR Sb. 10 217–243.) · Zbl 0202.11203 [10] Landim, C. (1993). Conservation of local equilibrium for attractive particle systems on $$Z^d$$. Ann. Probab. 21 1782–1808. JSTOR: · Zbl 0798.60085 · doi:10.1214/aop/1176989000 · links.jstor.org [11] Landim, C. (2003). Personal communication. [12] Liggett, T. M. (1978). Random invariant measures for Markov chains and independent particle systems. Z. Wahrsch. Verw. Gebiete 45 297–313. · Zbl 0373.60076 · doi:10.1007/BF00537539 [13] Liggett, T. M. (1985). Interacting Particle Systems . Springer, New York. · Zbl 0559.60078 [14] Liggett, T. M. (1999). Stochastic Interacting Systems : Contact , Voter and Exclusion Processes . Springer, Berlin. · Zbl 0949.60006 [15] Oleinik, O. A. (1957). Discontinuous solutions of nonlinear differential equations. Uspekhi Mat. Nauk (N.S.) 12 3–73. [English translation in Amer. Math. Soc. Transl. Ser. ( 2 ) 26 (1963) 95–172.] [16] Rezakhanlou, F. (1991). Hydrodynamic limit for attractive particle systems on $$Z^d$$. Comm. Math. Phys. 140 417–448. · Zbl 0738.60098 · doi:10.1007/BF02099130 [17] Smoller, J. (1983). Shock Waves and Reaction–Diffusion Equations . Springer, New York. · Zbl 0508.35002 [18] Spitzer, F. (1970). Interaction of Markov processes. Adv. Math. 5 246–290. · Zbl 0312.60060 · doi:10.1016/0001-8708(70)90034-4
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.