Brownian motion and Hardy’s inequality in \(L^ 2\). (Mouvement brownien et inégalité de Hardy dans \(L^ 2\).) (French) Zbl 0739.60073

Donati-Martin, C.; Yor, M.

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

Keywords:

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