Summary
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.
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Norris, J.R., Rogers, L.C.G. & Williams, D. Self-avoiding random walk: A Brownian motion model with local time drift. Probab. Th. Rel. Fields 74, 271–287 (1987). https://doi.org/10.1007/BF00569993
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DOI: https://doi.org/10.1007/BF00569993