I know that many things are known when dealing with random walks on a finite (or even infinite) graph: mixing time, returns to origin, etc. All is based in the use of the Markovian property of such a random walk (I am assuming that on each vertex we can choose the next one among neighbours with uniform probability distribution).

My question is the following: when dealing with non-backtracking random walks (namely, we cannot go back through an edge we have just used, inducing on every edge a uniform distribution) we lose all markovian property, but this can be manage by taking orientation on edges.

Q: is there some kind of 'universal' result for regular graphs concerning the existence of stationary distribution for such random walks?

I have looked for bibliography on this topic (this should be like the first question on these models), but I have not been able to find any reference on this.