Sur la régularité de certaines fonctions aléatoires d’Ornstein- Uhlenbeck. (On the regularity of certain Ornstein-Uhlenbeck random processes). (French) Zbl 0706.60041

Fernique, X.

Ann. Inst. Henri Poincaré, Probab. Stat. 26, No. 3, 399-417 (1990).

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

MSC:

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

Keywords:

sample path continuity; Ornstein-Uhlenbeck processes; calculus of variations