Continuity of \(\ell ^ 2\)-valued Ornstein-Uhlenbeck processes
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.
scientific article; zbMATH DE number 4147233