A symmetry property for a class of random walks in stationary random environments on \(\mathbb Z\) (Q2897146)

Random walks \((S_n)_{n \geq 0}\) and \((S^*_n)_{n \geq 0}\) on \(\mathbb{Z}\), described by dual Markov chains, are considered. Their transition probabilities \(\Pr(S_{n+1}=y+z|S_{n}=y)=p_{y,y+z}\), \(\Pr(S^*_{n+1}=y+z|S^*_{n}=y)=p^*_{y,y+z}\) are related by condition \(p^*_{y,y+z}=p_{y,y-z}\) for all \(y\), \(z\). If all \(p_{y,y+z}\), \(p^*_{y,y+z}\) do not depend on \(y\) then obviously \(\Pr(S^*_{n}=-x|S^*_{n}=0)=\Pr(S_{n}=x|S_{n}=0)\), but this equality does not hold in general case.NEWLINENEWLINEIn the article this equality is established in average (annealed) sense for dual random walks with two transition jumps in a stationary random environment. As the main tool the correspondence formula between the laws \(\Pr(S^*_{n}|S^*_{0})\), \(\Pr(S_{n}|S_{0})\) is used. In the case of more than two jumps the above equality is not true. A counterexample is given.