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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
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.

##### MSC:
 60G17 Sample path properties 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60J65 Brownian motion