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Regularity of $$\ell$$ 2-valued Ornstein-Uhlenbeck processes. (English) Zbl 0645.60046
Let $$X_ i(t)$$, $$i=1$$,..., $$t\geq 0$$, be a family of independent one- dimensional and stationary Ornstein-Uhlenbeck processes with drift and diffusion coefficients $$(-\lambda_ i)$$ and $$(\delta_ i)$$, respectively. Two properties of this family are proved.
1) Let $$X(t)=(X_ i(t))$$ be an $$\ell$$ 2-valued stochastic process. Then the process X(t) is continuous in $$\ell$$ 2 provided $$\sum \sigma$$ $$4_ i/\lambda_ i<\infty$$. 2) Let for every fixed $$t\geq 0$$, $$X_ i(t)$$ tend to zero with probability one. Then there exists $$x_ 0\geq 0$$ such that for some $$t\geq 0$$ the sequence $$X_ i(t)$$ clusters at each point of the interval $$[-x_ 0$$, $$x_ 0]$$ with probability one.
Reviewer: B.Goldys

##### MSC:
 60G17 Sample path properties 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60G15 Gaussian processes 60J60 Diffusion processes