Topics in self-interacting random walks

Abstract

The principle focus of this thesis is self-interacting random walks. A self-interacting random walk is a walk on a graph with its past influencing its future. In contrast to the regular random walks, self-interacting random walks are genuinely non-Markovian. Correspondingly, most of the standard tools of the theory of random walks are not directly available for the analysis of these models. Typically, this requires a significant adjustment and novel ad-hoc approaches in order to be applied. In this thesis we study two such processes, namely, excited random walks (ERWs) and directionally reinforced random walks (DRRWs). ERWs have actively attracted many mathematicians in recent years, and several basic questions regarding these random walks on Zd and trees have been answered. Nonetheless, despite all the effort done of late, there are still fundamental questions about ERWs to be answered. Here, we consider a transient ERW on Z and study the asymptotic behavior of the occupation time of a currently most visited site. In particular, our results imply that, in contrast to the random walks in random environment, a transient excited random walk does not spend an asymptotically positive fraction of time at its favorite (most visited up to a date) sites. DRRWs were originally introduced by Mauldin, Monticino, and von Weizsacker. In this thesis, we consider a generalized version of these processes and obtain a stable limit theorem for the position of the random walk in higher dimensions. This extends a result of Horvath and Shao that was previously obtained in dimension one only.