an:04147233
Zbl 0699.60052
Iscoe, I.; Marcus, M. B.; McDonald, D.; Talagrand, M.; Zinn, J.
Continuity of \(\ell ^ 2\)-valued Ornstein-Uhlenbeck processes
EN
Ann. Probab. 18, No. 1, 68-84 (1990).
00155590
1990
j
60H20 60G15 60G17
stochastic evolution equations; almost-sure continuity; Ornstein- Uhlenbeck process
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.
R.Curtain