We consider random walks in Dirichlet environment (RWDE) on $\Z ^d$, for $ d \geq 3 $, in the sub-ballistic case. We associate to any parameter $ (\alpha_1, \dots, \alpha _{2d}) $ of the Dirichlet law a time-change to accelerate the walk. We prove that the continuous-time accelerated walk has an absolutely continuous invariant probability measure for the environment viewed from the particle. This allows to characterize directional transience for the initial RWDE. It solves as a corollary the problem of Kalikow's $0-1$ law in the Dirichlet case in any dimension. Furthermore, we find the polynomial order of the magnitude of the original walk's displacement

We sharpen ellipticity criteria for random walks in i.i.d. random environments introduced by Campos and Ramírez which ensure ballistic behavior. Furthermore, we construct new examples of random environments for which the walk satisfies the polynomial ballisticity criteria of Berger, Drewitz and Ramírez. As a corollary, we can exhibit a new range of values for the parameters of Dirichlet random environments in dimension d = 2 under which the corresponding random walk is ballistic.Keywords: Dirichlet distribution; max-flow min-cut theorem; random walk in random environment; reinforced random walks This is an electronic reprint of the original article published by the ISI/BS in Bernoulli, 2016, Vol. 22, No. 2, 969-994. This reprint differs from the original in pagination and typographic detail. 1350-7265 c 2016 ISI/BS 2É. Bouchet, A.F. Ramírez and C. Sabot probabilities P x,ω (X n+1 = x + e|X n = x) = ω(x, e)for each x ∈ Z d and e ∈ U .Let P be a probability measure defined on the environment space Ω endowed with its Borel σ-algebra, such that {ω(x): x ∈ Z d } is i.i.d. under P. We call P x,ω the quenched law of the random walk in random environment (RWRE) starting from x, and P x := P x,ω dP(ω) the averaged or annealed law of the RWRE starting from x. The law P is said to be elliptic if for every x ∈ Z d and e ∈ U , P(ω(x, e) > 0) = 1. We say that P is uniformly elliptic if there exists a constant γ > 0 such that for every x ∈ Z d and e ∈ U , P(ω(x, e) ≥ γ) = 1.Given l ∈ S d−1 we say that the RWRE is transient in direction l if P 0 (A l ) = 1, with A l := lim n→∞ X n · l = ∞ .Furthermore, it is ballistic in direction l if P 0 -a.s. lim inf n→∞

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 Seppäläinen in [9] and Berger and Zeitouni in [2] under the assumption of large finite moments for the regeneration time. In this paper, with the extra (T )γ condition of Sznitman we reduce the moment condition to E(τ 2 (ln τ ) 1+m ) < +∞ for m > 1 + 1/γ, which allows the inclusion of new non-uniformly elliptic examples such as Dirichlet random environments.

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