Temps de sejour et oscillation du mouvement brownien au voisinage de la sphère euclidienne. (French) Zbl 0541.60028

Goldman, A.

Ann. Probab. 12, 829-842 (1984).

Summary: 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 \[ B_ f(a,\omega)=\int^{+\infty}_{0}1_{B(0,a)}(\omega(t))f(F(a,t,\omega)/F (a,+\infty,\omega))dt, \] where \(f:(0,1)\to R^+\) is an arbitrary bounded measurable function, coincides with the limiting distribution, when \(n\to +\infty\), of the weighted sojourn time \(n^{-1}T_ f(a\sqrt{n},\omega)=n^{- 1}\sum^{+\infty}_{k=0}1_{B(0,a\sqrt{n})}(S_ k(\omega))f((n(a\sqrt{n},k,\omega)/(n(a\sqrt{n},+\infty,\omega))\) for a standard random walk in \(Z^ p\) where n(b,k,\(\omega)\) denote the number of crossings 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 \(t_ a(\omega)\).

MSC:

60F17	Functional limit theorems; invariance principles
60G50	Sums of independent random variables; random walks
60J65	Brownian motion
33C10	Bessel and Airy functions, cylinder functions, \({}_0F_1\)
60J55	Local time and additive functionals
60G17	Sample path properties
60G60	Random fields

Keywords:

oscillation; sojourn time