Self-Interacting Random Walks

Let \({\mu }_1,…{\mu }_k\) be \(d\)-dimensional probability measures in \(\R^d\) with mean 0. At each step we choose one of the measures based on the history of the process and take a step according to that measure. We give conditions for transience of such processes and also construct examples of recurrent processes of this type. In particular, in dimension 3 we give the complete picture: every walk generated by two measures is transient and there exists a recurrent walk generated by three measures.