Integrability of exit times and ballisticity for random walks in Dirichlet environment. (English) Zbl 1192.60113

Tournier, Laurent

Electron. J. Probab. 14, 431-451 (2009).

Summary: We consider random walks in Dirichlet random environment. Since the Dirichlet distribution is not uniformly elliptic, the annealed integrability of the exit time out of a given finite subset is a non-trivial question. In this paper we provide a simple and explicit equivalent condition for the integrability of Green functions and exit times on any finite directed graph. The proof relies on a quotienting procedure allowing for an induction argument on the cardinality of the graph. This integrability problem arises in the definition of Kalikow auxiliary random walk. Using a particular case of our condition, we prove a refined version of the ballisticity criterion given by N. Enriquez and C. Sabot [C. R., Math., Acad. Sci. Paris 335, No. 11, 941–946 (2002; Zbl 1016.60051)].

MSC:

60K37	Processes in random environments
60J10	Markov chains (discrete-time Markov processes on discrete state spaces)

Keywords:

random walks in random environment; Dirichlet distribution; exit time; reinforced random walks; quotient graph; ballisticity

Citations:

Zbl 1016.60051