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

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
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