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Zbl 0739.60073
Donati-Martin, C.; Yor, M.
Brownian motion and Hardy's inequality in $L\sp 2$. (Mouvement brownien et inégalité de Hardy dans $L\sp 2$.)
(French)
[A] Séminaire de probabilités XXIII, Lect. Notes Math. 1372, 315-323 (1989).

[For the entire collection see Zbl 0722.00030.]\par This note is concerned with the convergence of certain indefinite integrals associated with the real Brownian motion $B=(B\sb s: s\ge 0)$ and the Ornstein-Uhlenbeck process $Y=(Y\sb s: s\ge 0)$. Specifically, the authors show:\par (1) If $\phi\in L\sp 1\sb{loc}((0,1],dx)$, then $\lim\sb{\varepsilon\downarrow 0}\int\sp 1\sb \varepsilon\phi(u)B\sb udu$ exists in probability iff $\Phi\in L\sp 2((0,1])$ and $\lim\sb{\varepsilon\downarrow 0}\sqrt{\varepsilon}\Phi(\varepsilon)=0$, where $\Phi(u)=\int\sp 1\sb u\phi(s)ds$.\par (2) If $g\in L\sp 2([0,\infty),dx)$, then $\int\sp t\sb 0 g(s)Y\sb s ds$ converges a.s. and in $L\sp 2$ as $t\to \infty$.\par (3) If $g\in L\sp 2([0,\infty),dx)$ and $\mu\ne 0$, then $\int\sp t\sb 0 g(s)e\sp{i\mu B\sb s}ds$ converges a.s. and in $L\sp 2$ as $t\to \infty$.\par Moreover, (3) is extended to symmetric Lévy processes. The motivation for this study comes from a link between Brownian motion and Hardy's $L\sp 2$ inequality.
[J.Bertoin (Paris)]
MSC 2000:
*60J65 Brownian motion
60E15 Inequalities in probability theory

Keywords: Hardy inequality; indefinite integrals; Brownian motion; Ornstein- Uhlenbeck process; symmetric Lévy processes

Citations: Zbl 0722.00030

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