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Brownian motion and Hardy’s inequality in $$L^ 2$$. (Mouvement brownien et inégalité de Hardy dans $$L^ 2$$.) (French) Zbl 0739.60073
Séminaire de probabilités XXIII, Lect. Notes Math. 1372, 315-323 (1989).
[For the entire collection see Zbl 0722.00030.]
This note is concerned with the convergence of certain indefinite integrals associated with the real Brownian motion $$B=(B_ s: s\geq 0)$$ and the Ornstein-Uhlenbeck process $$Y=(Y_ s: s\geq 0)$$. Specifically, the authors show:
(1) If $$\phi\in L^ 1_{loc}((0,1],dx)$$, then $$\lim_{\varepsilon\downarrow 0}\int^ 1_ \varepsilon\phi(u)B_ udu$$ exists in probability iff $$\Phi\in L^ 2((0,1])$$ and $$\lim_{\varepsilon\downarrow 0}\sqrt{\varepsilon}\Phi(\varepsilon)=0$$, where $$\Phi(u)=\int^ 1_ u\phi(s)ds$$.
(2) If $$g\in L^ 2([0,\infty),dx)$$, then $$\int^ t_ 0 g(s)Y_ s ds$$ converges a.s. and in $$L^ 2$$ as $$t\to \infty$$.
(3) If $$g\in L^ 2([0,\infty),dx)$$ and $$\mu\neq 0$$, then $$\int^ t_ 0 g(s)e^{i\mu B_ s}ds$$ converges a.s. and in $$L^ 2$$ as $$t\to \infty$$.
Moreover, (3) is extended to symmetric Lévy processes. The motivation for this study comes from a link between Brownian motion and Hardy’s $$L^ 2$$ inequality.
Reviewer: J.Bertoin (Paris)

##### MSC:
 60J65 Brownian motion 60E15 Inequalities; stochastic orderings