Excited Brownian motions as limits of excited random walks
Résumé
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 $\varphi$, of its local time. It was introduced by Norris, Rogers and Williams few years ago, 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 Tóth and his coauthors in multiple occasions, which goes through Ray-Knight type results. Namely we first prove, when $\varphi$ is bounded and sufficiently regular, that the convergence holds at the level of the local time processes. Then, if we assume in addition that $\varphi$ is nonnegative and that its integral is strictly lower than $1$, we can deduce the convergence at the level of the full processes.
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