Title: Convergence and non-convergence of some self-interacting random walks to Brownian motion perturbed at extrema

Abstract: Generalized Ray-Knight theorems for edge local times proved to be a very useful tool for studying the limiting behavior of several classes of self-interacting random walks (SIRWs) on integers. Examples include some reinforced random walks, excited random walks, rotor walks with defects. I shall describe two classes of SIRWs introduced and studied by Balint Toth (1996), asymptotically free and polynomially self-repelling SIRWs, and discuss new results which resolve an open question posed in Toth’s paper. We show that in the asymptotically free case the rescaled SIRWs converge to a perturbed Brownian motion (conjectured by Toth) while in the polynomially self-repelling case the convergence to the conjectured process fails in spite of the fact that generalized Ray-Knight theorems clearly identify the unique candidate in the class of all perturbed Brownian motions. This negative result was somewhat unexpected. The question whether there is convergence in the polynomially self-repelling case and, if yes, then how to describe the limiting process is open. 
*This is joint work with Thomas Mountford, EPFL, and Jonathon Peterson, Purdue University.