Excited Brownian motions as limits of excited random walks

TL;DR

This paper proves that excited random walks, when properly rescaled, converge in distribution to an excited Brownian motion, a process characterized by a local time-dependent drift, extending previous theoretical frameworks.

Contribution

It establishes the convergence of excited random walks to excited Brownian motions using a novel renormalization and analytical approach based on Ray-Knight type results.

Findings

Convergence in law of excited random walks to excited Brownian motion.

Identification of the limiting process as a semi-martingale with local time-dependent drift.

Methodology applicable to similar stochastic process convergence proofs.

Abstract

We obtain the convergence in law of a sequence of excited (also called cookies) random walks toward an excited Brownian motion. This last process is a continuous semi-martingale whose drift is a function, say $\phi$, of its local time. It was introduced by Norris, Rogers and Williams as a simplified version of Brownian polymers, and then recently further studied by the authors. To get our results we need to renormalize together the sequence of cookies, the time and the space in a convenient way. The proof follows a general approach already taken by T\'oth and his coauthors in multiple occasions, which goes through Ray-Knight type results. Namely we first prove, when $\phi$ is bounded and Lipschitz, that the convergence holds at the level of the local time processes. This is done via a careful study of the transition kernel of an auxiliary Markov chain which describes the local time at a given level. Then we prove a tightness result and deduce the convergence at the level of the full processes.