Random Dirichlet environment viewed from the particle in dimension $d\ge3$

Abstract

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 by a $2d$-tuple of positive reals called weights. In this paper, we characterize for $d\ge3$ the weights for which there exists an absolutely continuous invariant probability distribution for the process viewed from the particle. We can deduce from this result and from [Ann. Inst. Henri Poincaré Probab. Stat. 47 (2011) 1–8] a complete description of the ballistic regime for $d\ge3$.