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Continuity of $$\ell ^ 2$$-valued Ornstein-Uhlenbeck processes. (English) Zbl 0699.60052
The authors consider the almost-sure continuity of the following $$\ell^ 2$$-valued Ornstein-Uhlenbeck process given by $dX_ t=AX_ tdt+\sqrt{2\alpha}dB_ t,$ where A is a positive, self-adjoint operator on $$\ell^ 2$$, $$B_ t$$ is a cylindrical Brownian motion on $$\ell^ 2$$ and $$\alpha$$ is a positive diagonal operator on $$\ell^ 2$$. They give a simple sufficient condition for the almost-sure continuity of $$X_ t$$ in $$\ell_ 2$$ and show that it is quite sharp. Furthermore, they obtain necessary and sufficient conditions in special cases.
Reviewer: R.Curtain

##### MSC:
 60H20 Stochastic integral equations 60G15 Gaussian processes 60G17 Sample path properties
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