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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
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