Monotonicity and regularity of the speed for excited random walks in higher dimensions

Abstract

We introduce a method for studying monotonicity of the speed of excited random walks in high dimensions, based on a formula for the speed obtained via cut-times and Girsanov's transform. While the method gives rise to similar results as have been or can be obtained via the expansion method of van der Hofstad and Holmes, it may be more palatable to a general probabilistic audience. We also revisit the law of large numbers for stationary cookie environments. In particular, we introduce a new notion of $\mathcal{e}_1 -$ exchangeable cookie environment and prove the law of large numbers for this case.