Construction of the 1d Self-repelling Brownian Polymer

Abstract: We consider the self-repelling Brownian polymer, introduced in [APP83], which is formally defined as the solution of a singular SDE. The singularity comes from the drift term, which is given by the negative gradient of the local time. We construct a solution of the equation in d = 1, give a dynamic characterisation of its law, and show that it is the limiting distribution for a natural family of approximations. In addition, we show that the solution is superdiffusive. As part of the construction, we consider a singular SPDE which is solved by the (recentered) local time. Using the method of energy solutions, we show that this SPDE is well-posed, and prove that this property can be transferred to the original process. Our results hold for a larger class of drift terms, in which the gradient of the local time is a special case.