Tubular recurrence (Q1410401)

Let \(G =(V, E)\) with vertex set \(V\) and edge set \(E\) be a connected locally finite graph with all edges directed. The author introduces the directed edge-reinforced random walk on \(G\); each edge is assigned a strictly positive real number as initial weight. In each step the object executing the random walk moves from one vertex to another adjacent one with a probability proportional to the weight of the edge chosen. Each time an edge is traversed by the object the weight is increased by unity. In a similar way an environment is defined as a function defined on the collection of all the edges; if \((u, v)\) is an edge from vertex \(u\) to \(v\), then \(0 \leq \omega (u,v) \leq 1\) with the property \( \sum \omega (u,v)=1 \text{for all} u \in V,\) where the summation is over \(\{v \in V: (u, v)\) is an edge\}. By introducing a Markov chain on \(G\) induced by the environment, the author creates a Markov chain with \(\omega\) dependent Markov transition probabilities. If at this stage a probability measure is introduced on \(\Omega\) the collection of all environments, a random walk in a random environment is realized. The main result of the paper is that a directed edge reinforced random walk is equivalent to a random walk in a random environment that is independent of the random walk.