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

Abstract

A natural model for a ‘self-avoiding’ Brownian motion inR d, when specialised and simplified tod=1, becomes the stochastic differential equation $$X_t = B_t - \int\limits_0^t g (X_s ,L(s,X_s ))ds$$ , where {L(t, x):t≧0,x∈R} is the local time process ofX. ThoughX 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 $$\mathop {\lim }\limits_{t \to \infty } X_t /t$$ exists almost surely, and to identify the limit.