Ray-Knight Theorem Research Articles

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

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.

Some connections between excursion theory and the discrete Schrödinger equation with random potentials

We use here discrete excursion theory, and the Ray-Knight theorem, in order to study questions concerning the analyticity of the density of states at “low disorder”, as well as its smoothness properties, in one dimension.

The most visited site of Brownian motion and simple random walk

Let L(t, x) be the local time at x for Brownian motion and for each t, let $$\bar V(t) = \inf \{ x\underline{\underline > } 0;L(t,x) \vee L(t, - x) = \mathop {\sup }\limits_y L(t,y)\} $$ , the absolute value of the most visited site for Brownian motion up to time t. In this paper we prove that ¯V(t) is transient and obtain upper and lower bounds for the rate of growth of ¯V(t). The main tools used are the Ray-Knight theorems and William's path decomposition of a diffusion. An invariance principle is used to get analogous results for simple random walks. We also obtain a law of the iterated logarithm for ¯V(t).