A clever (self-repelling) burglar

Abstract

We derive the following property of the "true self-repelling motion", a continuous real-valued self-interacting process $(X_t, t \ge 0)$ introduced by Balint Toth and Wendelin Werner. Conditionally on its occupation time measure at time one (which is the information about how much time it has spent where before time one), the law of $X_1$ is uniform in a certain admissible interval. This contrasts with the corresponding conditional distribution for Brownian motion that had been studied by Warren and Yor.