Self-avoiding random walk: A Brownian motion model with local time drift

Abstract

A natural model for a 'self-avoiding' Brownian motion in Rd, when specialised and simplified to d=1, becomes the stochastic differential equation {Mathematical expression}, where {L(t, x):t≧0, x∈R} is the local time process of X. Though X is not Markovian, an analogue of the Ray-Knight theorem holds for {L(∞, x):x∈R}, which allows one to prove in many cases of interest that {Mathematical expression} exists almost surely, and to identify the limit. © 1987 Springer-Verlag.