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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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