A clever (self-repelling) burglar

TL;DR

This paper investigates the true self-repelling motion, a self-interacting process, revealing that conditioned on its occupation time at a fixed time, its position is uniformly distributed within an interval, contrasting with Brownian motion.

Contribution

It establishes a novel property of the true self-repelling motion, showing its conditional distribution given occupation time is uniform, unlike Brownian motion.

Findings

Conditional distribution of X_1 is uniform within an interval.

Contrasts with the behavior of Brownian motion.

Provides new insights into self-interacting stochastic processes.

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.