We prove a quenched central limit theorem for random walks in i.i.d. weakly elliptic random environments in the ballistic regime. Such theorems have been proved recently by Rassoul-Agha and Seppalainen in [10] and Berger and Zeitouni in [2] under the
assumption of large finite moments for the regeneration time. In this paper, with the extra $(T)_{gamma}$ condition of Sznitman we reduce the moment condition to ${Bbb E}(tau^2(ln tau)^{1+m})<+infty$ for $m>1+1/gamma$, which allows the inclusion of new non-uniformly elliptic examples such as Dirichlet random environments.
We consider random walks in random Dirichlet environment (RWDE) which is a special type of random walks in random environment where the exit probabilities at each site are i.i.d. Dirichlet random variables. On ${mathbb Z}^d$, RWDE are parameterized b
y a 2d-uplet of positive reals called weights. In this paper, we characterize for $dge 3$ the weights for which there exists an absolutely continuous invariant probability for the process viewed from the particle. We can deduce from this result and from [27] a complete description of the ballistic regime for $dge 3$.
We consider random walks in random Dirichlet environment (RWDE) which is a special type of random walks in random environment where the exit probabilities at each site are i.i.d. Dirichlet random variables. On $Z^d$, RWDE are parameterized by a $2d$-
uplet of positive reals. We prove that for all values of the parameters, RWDE are transient in dimension $dge 3$. We also prove that the Green function has some finite moments and we characterize the finite moments. Our result is more general and applies for example to finitely generated symmetric transient Cayley graphs. In terms of reinforced random walks it implies that directed edge reinforced random walks are transient for $dge 3$.
We consider random walks in a random environment that is given by i.i.d. Dirichlet distributions at each vertex of Z^d or, equivalently, oriented edge reinforced random walks on Z^d. The parameters of the distribution are a 2d-uplet of positive real
numbers indexed by the unit vectors of Z^d. We prove that, as soon as these weights are nonsymmetric, the random walk in this random environment is transient in a direction with positive probability. In dimension 2, this result can be strenghened to an almost sure directional transience thanks to the 0-1 law from [ZM01]. Our proof relies on the property of stability of Dirichlet environment by time reversal proved in [Sa09]. In a first part of this paper, we also give a probabilistic proof of this property as an alternative to the change of variable computation used in that article.
We consider transient one-dimensional random walks in random environment with zero asymptotic speed. An aging phenomenon involving the generalized Arcsine law is proved using the localization of the walk at the foot of valleys of height $log t$. In t
he quenched setting, we also sharply estimate the distribution of the walk at time $t$.
We show how a description of Brownian exponential functionals as a renewal series gives access to the law of the hitting time of a square-root boundary by a Bessel process. This extends classical results by Breiman and Shepp, concerning Brownian moti
on, and recovers by different means, extensions for Bessel processes, obtained independently by Delong and Yor.
