Random walk in slowly changing environments

A Random Walk in Changing Environment (RWCE) is a weighted random walk on a locally finite, connected graph G with random, time-dependent edge-weights. This includes self-interacting random walks, where the edge-weights depend on the history of the process. In general, even the basic question of recurrence or transience for RWCEs is difficult, especially when the underlying graph contains cycles. In this note, we derive a condition for recurrence or transience that is too restrictive for classical RWCEs but instead works for any graph G. Namely, we show that any bounded RWCE on G with “slowly” changing edge-weights inherits the recurrence or transience of the initial weighted graph.