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**Characterization of stationary measures for one-dimensional exclusion processes.**
*(English)*
Zbl 1039.60086

The authors provide a characterization of stationary measures for one-dimensional exclusion processes in adding to known results their new results as follows. The one-dimensional exclusion process with random walk kernel is a continuous time Markov process on \(\{0,1\}^{\mathbb{Z}} \). Particles can only jump on empty sites, and this jump probability is given by the random walk kernel \( p \). If the random walk kernel is irreducible, it is well known that the Bernoulli product measures are the extremal translation invariant stationary measures for the process. When the mean \(\mu:=\sum_{x}xp(x) =0 \), there are no non-translation invariant stationary measures. Stationary measures that are not translation invariant are known to exist for finite range \( p \) with positive mean (i.e. there is a bias to the right). These so-called profile/blocking measures have particle densities that tend to \( 1 \) as \( x\to\infty \) and tend to \( 0 \) as \( x\to -\infty \).

The first main result of the article goes in the opposite direction, i.e., the authors show that, when \( p \) is irreducible with positive mean, the only possible extremal non translation invariant stationary measures consist of a profile measure together with its translates. The second result shows that for \(p \) with finite mean but \(\sum_{x<0}x^2p(x)=\infty \) any non translation invariant stationary measure is not a blocking measure. The authors also present two results on the existence of blocking measures under the monotonicity condition on \(p \). In particular they substantially relax the tail behaviour assumed by M. Bramson and T. Mountford [Ann. Probab. 30, 1082–1130 (2002; Zbl 1042.60062)] when they assume slightly more than three moments on the left tail. In this case stationary blocking measures do exist. The last result gives the existence of a stationary blocking measure when \(p \) has finite strict positive mean with \( p(y)\leq p(x) \) and \( p(-y)\leq p(-x) \) for \( 1\leq x\leq y \) and \(p(-x)\leq p(x) \) for \(x\geq 1 \) and \(\sum_{x<0}x^2p(x)<\infty \).

The first main result of the article goes in the opposite direction, i.e., the authors show that, when \( p \) is irreducible with positive mean, the only possible extremal non translation invariant stationary measures consist of a profile measure together with its translates. The second result shows that for \(p \) with finite mean but \(\sum_{x<0}x^2p(x)=\infty \) any non translation invariant stationary measure is not a blocking measure. The authors also present two results on the existence of blocking measures under the monotonicity condition on \(p \). In particular they substantially relax the tail behaviour assumed by M. Bramson and T. Mountford [Ann. Probab. 30, 1082–1130 (2002; Zbl 1042.60062)] when they assume slightly more than three moments on the left tail. In this case stationary blocking measures do exist. The last result gives the existence of a stationary blocking measure when \(p \) has finite strict positive mean with \( p(y)\leq p(x) \) and \( p(-y)\leq p(-x) \) for \( 1\leq x\leq y \) and \(p(-x)\leq p(x) \) for \(x\geq 1 \) and \(\sum_{x<0}x^2p(x)<\infty \).

Reviewer: Stefan Adams (Berlin)

### MSC:

60K35 | Interacting random processes; statistical mechanics type models; percolation theory |

60J25 | Continuous-time Markov processes on general state spaces |

82C22 | Interacting particle systems in time-dependent statistical mechanics |

### Citations:

Zbl 1042.60062
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\textit{M. Bramson} et al., Ann. Probab. 30, No. 4, 1539--1575 (2002; Zbl 1039.60086)

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### References:

[1] | BRAMSON, M. and MOUNTFORD, T. (2002). Stationary blocking measures for one-dimensional nonzero mean exclusion processes. Ann. Probab. 30 1082-1130. · Zbl 1042.60062 |

[2] | FERRARI, P. A., LEBOWITZ, J. L. and SPEER, E. (2001). Blocking measures for asy mmetric exclusion processes via coupling. Bernoulli 7 935-950. · Zbl 1002.60100 |

[3] | HOLLEY, R. and LIGGETT, T. M. (1978). The survival of contact processes. Ann. Probab. 6 198-206. · Zbl 0375.60111 |

[4] | LIGGETT, T. M. (1976). Coupling the simple exclusion process. Ann. Probab. 4 339-356. · Zbl 0339.60091 |

[5] | LIGGETT, T. M. (1977). Ergodic theorems for the asy mmetric simple exclusion process II. Ann. Probab. 5 795-801. · Zbl 0378.60104 |

[6] | LIGGETT, T. M. (1985). Interacting Particle Sy stems. Springer, Berlin. |

[7] | LIGGETT, T. M. (1999). Stochastic Interacting Sy stems: Contact, Voter and Exclusion Processes. Springer, Berlin. · Zbl 0949.60006 |

[8] | REZAKHANLOU, F. (1991). Hy drody namic limit for attractive particle sy stems on Zd. Comm. Math. physics 140 417-448. · Zbl 0738.60098 |

[9] | MINNEAPOLIS, MINNESOTA 55455 E-MAIL: bramson@math.umn.edu DEPARTMENT OF MATHEMATICS UNIVERSITY OF CALIFORNIA, LOS ANGELES 405 HILGARD AVENUE LOS ANGELES, CALIFORNIA 90095 E-MAIL: tml@math.ucla.edu malloy@math.ucla.edu |

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