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

Fernique, Xavier

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.

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