Transition probabilities of step-reinforced random walks

Abstract

The step-reinforced random walk (SRRW) either repeats a uniformly chosen past step or takes a fresh independent step. We consider a generalized SRRW on groups, where the selected past step is transformed by a random map. The transformations are independent of the fresh steps but may be dependent on one another. For every reinforcement parameter $α<1$, we obtain upper bounds on transition probabilities in terms of the geometry of the group. On $\mathbb{R}^d$, a non-singular step distribution yields bounds of order $n^{-d/2}$ for balls of any fixed radius, uniformly over their centers and without any moment assumption, and hence transience for $d\geq 3$. On finitely generated groups, we give bounds in terms of the isoperimetric profile. We also prove exponential decay for the elephant random walk on every nonamenable Cayley graph with a finite symmetric generating set and every memory parameter $p<1$. In particular, this answers a question of Mukherjee for Cayley trees.