True self-repelling motion above a general barrier

The true self-repelling motion is a continuous-time random process which was introduced by Tóth and Werner in 1998 to be a limit for the "true" self-avoiding random walk defined by Tóth 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 Tóth and Werner to this more general case is surprisingly difficult, especially when the barrier is irregular.