True self-repelling motion above a general barrier

TL;DR

This paper extends the construction of true self-repelling motion to general barriers, involving complex coalescing Brownian motions reflected and absorbed on irregular barriers, advancing understanding of self-interacting stochastic processes.

Contribution

It introduces a generalized framework for true self-repelling motion with arbitrary barriers, overcoming significant technical challenges in irregular cases.

Findings

Constructed coalescing Brownian motions for general barriers.

Extended the true self-repelling motion to irregular barriers.

Demonstrated the complexity of proofs in the generalized setting.

Abstract

The true self-repelling motion is a continuous-time random process which was introduced by T\'oth and Werner in 1998 to be a limit for the "true" self-avoiding random walk defined by T\'oth in 1995. The construction of the true self-repelling motion involves an uncountable system of coalescing Brownian motions starting from all points of the upper half-plane, related to the Brownian web, but reflected and absorbed on a "barrier" which is the abscissa axis. In this work, we consider much more general barriers, construct an uncountable system of coalescing Brownian motions reflected and absorbed on these barriers, and the true self-repelling motion associated to it. The extension of the proofs of T\'oth and Werner to this more general case is surprisingly difficult, especially when the barrier is irregular.