The Annals of Probability

Temps de Sejour et Oscillation du Mouvement Brownien au Voisinage de la Sphere Euclidienne

A. Goldman

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For a standard Brownian motion $\omega(t)$ in $R^p, p \geq 3$, let $t_a(\omega)$ be the last exit time from the ball $B(0, a)$ of radius $a$ centered at the origin and let $F(a, t, \omega)$ be the oscillation in the neighbourhood of sphere $S(0, a)$. The distribution of the functional $B_f(a, \omega) = \int^{+\infty}_0 1_{B(0,a)}(\omega(t))f\big(\frac{F(a, t, \omega)}{F(a, +\infty, \omega)}\big) dt,$ where $f: (0, 1) \rightarrow R^+$ is an arbitrary bounded measurable function, coincides with the limiting distribution, when $n \rightarrow +\infty$, of the weighted sojourn time $\frac{T_f(a\sqrt n, \omega)}{n} = \sum^{+\infty}_{k=0} 1_{B(0,a\sqrt n)}(S_k(\omega))f\big(\frac{n(a\sqrt n, k, \omega)}{n(a\sqrt n, +\infty, \omega)}\big)\big/n$ for a standard random walk in $Z^p$ where $n(b, k, \omega)$ denote the number of crossing $S(0, b)$ during the first $k$ steps. We give explicit formulas, in terms of Laplace transform, for the joint distribution of $B_f(a, \omega), F(a, +\infty, \omega)$ and $t_a(\omega)$.

Article information

Ann. Probab., Volume 12, Number 3 (1984), 829-842.

First available in Project Euclid: 19 April 2007

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Primary: 60F17: Functional limit theorems; invariance principles
Secondary: 60J15 60J65: Brownian motion [See also 58J65] 33A40 60J55: Local time and additive functionals 60G17: Sample path properties 60G60: Random fields

Brownian motion random walk sojourn time and oscillation Bessel functions Hausforff measure


Goldman, A. Temps de Sejour et Oscillation du Mouvement Brownien au Voisinage de la Sphere Euclidienne. Ann. Probab. 12 (1984), no. 3, 829--842. doi:10.1214/aop/1176993231.

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