Scaling Limit of Asymptotically-free Self-interacting Random Walks to Brownian Motion Perturbed at Extrema

We show convergence of a family of one-dimensional self-interacting random walks to Brownian motion perturbed at extrema under the diffusive scaling. This completes the functional limit theorem in KMP23 for the asymptotically free case when 0<p 12. The approach is to approximate the total drift experienced by the walker via analyzing directed edge local times, described by the branching-like processes. The analysis depends on the diffusion approximation of the branching-like processes obtained in the Ray-Knight type framework.