  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.
