##
**Quenched invariance principle for multidimensional ballistic random walk in a random environment with a forbidden direction.**
*(English)*
Zbl 1126.60090

The authors consider a random walk in random environment (RWRE) on \(\mathbb Z^d\), \(d\geq 2\), defined as follows. Let \(\omega =(\omega_x)_{x\in \mathbb Z^d}\), the random environment, be an i.i.d. set of probability vectors on \(\mathbb Z^d\); the law of \(\omega \) is denoted by \(\mathbb P\). Given \(\omega \), a random walk, \(X=(X_n)_{n \geq 0}\), starts at the origin, and, being located at \(x\), it jumps to \(y\) with probability \(\pi_{x,y}(\omega )\) which is defined by \(\omega_x=(\pi_{x,x+y})_{y\in \mathbb Z^d}\). The probability distribution \(P_0^\omega\) of \(X\) with fixed \(\omega \) is called quenched distribution. If the average over \(\omega \) is performed, the distribution is called annealed and denoted by \(P_0\). In the present paper the authors assume that the random walk is not allowed to retreat in a fixed direction \(\hat u\), that is \(\pi_{x,y}=0\) if \((y-x)\cdot \hat u<0\). This creates a drift in a certain direction and there is a law of large numbers with a non-zero speed \(v\). Moreover, it is assumed that the environment is non-nestling, elliptic, acts on more than two spatial dimensions and the step of the walk has \(2+\epsilon\) moment uniformly in the environment. The main result of the paper is a quenched invariance principle which states that the processes \( B_n=n^{-1/2}\{X_{[nt]}-[nt]v\}\) and \( \tilde B_n=n^{-1/2}\{X_{[nt]}-E_0^\omega(X[nt])\}\) converge to a Brownian motion with a certain diffusion matrix. The proof is based on previous papers of both authors [ALEA, Lat. Am. J. Probab. Math. Stat. 1, 111–147, electronic only (2006; Zbl 1115.60106)], where the annealed invariance principle was proved under similar conditions, and [Probab. Theory Relat. Fields 133, No. 3, 299–314 (2005; Zbl 1088.60094)], where a theorem allowing the proof of the quenched invariance principle under certain conditions was derived. The main difficulty in the paper is then verifying these conditions, in particular bounding the variance of the quenched mean: \(\mathbb E[| E^\omega_0(X_n)-E_0(X_n)| ^2]\leq C n^\gamma\), where \(\gamma \) depends on order of the moment that is assumed for the step distribution of \(X\).

Reviewer: Jiří Černý (Zürich)

### MSC:

60K37 | Processes in random environments |

60F17 | Functional limit theorems; invariance principles |

82D30 | Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses) |

### References:

[1] | Balázs, M., Rassoul-Agha, F. and Seppäläinen, T. (2006). The random average process and random walk in a space–time random environment in one dimension. Comm. Math. Phys. 266 499–545. · Zbl 1129.60097 · doi:10.1007/s00220-006-0036-y |

[2] | Bolthausen, E. and Sznitman, A.-S. (2002). On the static and dynamic points of view for certain random walks in random environment. Methods Appl. Anal. 9 345–375. · Zbl 1079.60079 · doi:10.4310/MAA.2002.v9.n3.a4 |

[3] | Bolthausen, E. and Sznitman, A.-S. (2002). Ten Lectures on Random Media . Birkhäuser, Basel. · Zbl 1075.60128 |

[4] | Burkholder, D. L. (1973). Distribution function inequalities for martingales. Ann. Probab. 1 19–42. · Zbl 0301.60035 · doi:10.1214/aop/1176997023 |

[5] | Derriennic, Y. and Lin, M. (2003). The central limit theorem for Markov chains started at a point. Probab. Theory Related Fields 125 73–76. · Zbl 1012.60028 · doi:10.1007/s004400200215 |

[6] | Durrett, R. (2004). Probability : Theory and Examples , 3rd ed. Brooks/Cole–Thomson, Belmont, CA. · Zbl 0709.60002 |

[7] | Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes . Wiley, New York. · Zbl 0592.60049 |

[8] | Kipnis, C. and Varadhan, S. R. S. (1986). Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions. Comm. Math. Phys. 104 1–19. · Zbl 0588.60058 · doi:10.1007/BF01210789 |

[9] | Maxwell, M. and Woodroofe, M. (2000). Central limit theorems for additive functionals of Markov chains. Ann. Probab. 28 713–724. · Zbl 1044.60014 · doi:10.1214/aop/1019160258 |

[10] | Rassoul-Agha, F. and Seppäläinen, T. (2005). An almost sure invariance principle for random walks in a space–time random environment. Probab. Theory Related Fields 133 299–314. · Zbl 1088.60094 · doi:10.1007/s00440-004-0424-1 |

[11] | Rassoul-Agha, F. and Seppäläinen, T. (2006). Ballistic random walk in a random environment with a forbidden direction. ALEA Lat. Am. J. Probab. Math. Stat. 1 111–147. · Zbl 1115.60106 |

[12] | Sznitman, A.-S. (2004). Topics in random walk in random environment. In Notes of the School and Conference on Probability Theory ( Trieste , 2002 ) . ICTP Lecture Series 203–266. Abdus Salam Int. Cent. Theoret. Phys., Trieste. · Zbl 1060.60102 |

[13] | Zeitouni, O. (2004). Random Walks in Random Environments . Springer, Berlin. · Zbl 1060.60103 |

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.