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Sur la régularité de certaines fonctions aléatoires d’Ornstein- Uhlenbeck. (On the regularity of certain Ornstein-Uhlenbeck random processes). (French) Zbl 0706.60041
Necessary and sufficient conditions for the a.s. sample path continuity of a certain class of Ornstein-Uhlenbeck processes (i.e., those for which an associated semigroup of operators is simultaneously diagonalizable) are given. In the special case of \(\ell_ p\)-continuity for separable solutions of the equation \[ dX=-\Lambda Xdt+\Sigma dW,\quad X(0)=0, \] where \(X(t)\in \ell_ p\), \(p\geq 2\), \(\Lambda\) diagonal \(\Sigma\) diagonal and non-negative, and W a Wiener process, these conditions are restated in a simpler form. Connections with some problems in the calculus of variations in \(\ell_ p\) appear in an appendix.
Reviewer: J.Mitro

60G15 Gaussian processes
60G17 Sample path properties
60B11 Probability theory on linear topological spaces
49J27 Existence theories for problems in abstract spaces
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