Random walk in random environment and their time-reversed counterpart

The random walk in Dirichlet environment is a random walk in random environment where the transition probabilities are independent Dirichlet random variables. This random walk exhibits a property of statistical invariance by time-reversal which leads to several results. More precisely, a time-reversed random walk in Dirichlet environment (with null divergence) is also a random walk in random environment where the transition probabilities are independent Dirichlet random variables with different parameters. We show that on all graphs that satisfy a few weak assumptions, a random walk in random environment with independent transition probabilities and such that the transition probabilities of the time-reversed random walk in random environment are also independent is a random walk in Dirichlet environment.