Abstract
In this paper we first obtain a necessary condition for \(H\)-cadlag modification and \(H\)-weakly cadlag modification of generalized Ornstein–Uhlenbeck processes with Levy noises in Hilbert spaces \(H\). Then we give a necessary and sufficient condition for the \(H\)-cadlag modification and \(H\)-weakly cadlag modification of Ornstein–Uhlenbeck processes driven by cylindrical \(\alpha \)-semistable processes. Secondly, we investigate the properties of cylindrical cadlag modification and \(V\)-cylindrical cadlag modification. Applying the obtained results to diagonal Ornstein–Uhlenbeck processes with \(\alpha \)-stable noises, we show a necessary and sufficient condition for cylindrical cadlag modification and \(V\)-cylindrical cadlag modification in the symmetric case for \(\alpha \in (0,1)\) and give a sufficient condition in the general case for \(\alpha \in (0,2)\). Some examples illustrate the relations among the concepts of various cadlag modifications.
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