Fernique, Xavier La régularité des fonctions aléatoires d’Ornstein-Uhlenbeck à valeurs dans \(\ell ^ 2\); le cas diagonal. (Continuity of \(\ell ^ 2\)- valued Ornstein-Uhlenbeck random functions; the diagonal case). (French) Zbl 0674.60040 C. R. Acad. Sci., Paris, Sér. I 309, No. 1, 59-62 (1989). Summary: We characterize the regularity of paths of \(\ell^ 2\)-valued solutions of the diagonal Langevin equation \(dV=-\Lambda Vdt+\Sigma dW\), \(t\in {\mathbb{R}}^+\), where \(\Lambda\) is diagonal positive, \(\Sigma\) is diagonal non-negative and W is a Wiener process with independent normalized components: their paths are continuous in \(\ell^ 2\) if and only if they are in this space and the integral \[ \int \log^+(\sup \{\lambda_ k:\quad \sigma^ 2_ k>\lambda_ kx\})dx \] is finite. Cited in 7 Documents MSC: 60G17 Sample path properties 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60J65 Brownian motion Keywords:Ornstein-Uhlenbeck random functions; regularity of paths; diagonal Langevin equation; Wiener process PDF BibTeX XML Cite \textit{X. Fernique}, C. R. Acad. Sci., Paris, Sér. I 309, No. 1, 59--62 (1989; Zbl 0674.60040)