an:04104104
Zbl 0674.60040
Fernique, Xavier
La r??gularit?? des fonctions al??atoires d'Ornstein-Uhlenbeck ?? valeurs dans \(\ell ^ 2\); le cas diagonal. (Continuity of \(\ell ^ 2\)- valued Ornstein-Uhlenbeck random functions; the diagonal case)
FR
C. R. Acad. Sci., Paris, S??r. I 309, No. 1, 59-62 (1989).
00171515
1989
j
60G17 60H10 60J65
Ornstein-Uhlenbeck random functions; regularity of paths; diagonal Langevin equation; Wiener process
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.