A Note on time regularity of generalized Ornstein–Uhlenbeck processes with cylindrical stable noise

Time Regularity of Generalized Ornstein–Uhlenbeck Processes with Lévy Noises in Hilbert Spaces

“…Proof of Theorem 2.1: The proof of part (1) was presented in [12]. We know that an H -càdlàg modification implies an H -weakly càdlàg modification; in general, the inverse direction does not hold, as the following example shows.…”

“…In [15], Peszat and Zabczyk explore the aforementioned related problems, obtain some interesting results, and develop some useful methods in connection with this topic. In the present paper, we continue the studies presented in [12] and [15] and focus on giving some criteria of the various càdlàg modifications of processes in Hilbert spaces.…”

“…Some criteria for càdlàg and non-càdlàg properties of X or L in a suitable space are given. In [12], we show a necessary and sufficient condition of H -valued processes having H -càdlàg modification, and, on the basis of this condition, we obtain three equivalent assertions of the H -càdlàg modification of OU processes with cylindrical α-stable processes.…”

“…(1.1) have been obtained (see [4,7,8,11,13] and references therein). Recently, some substantial results were obtained on the time regularity of SPDEs driven by a Lévy process (see [3,5,10,12,15,17,18] and reference therein). Some criteria for càdlàg and non-càdlàg properties of X or L in a suitable space are given.…”

Infinite dimensional Ornstein-Uhlenbeck processes driven by Lévy processes

“…In the case where the L n 's have symmetric α-stable distributions (0 < α < 2), a finer analysis may be carried out; in [45] the authors obtain a number of interesting properties of the solution, including gradient estimates on the transition semigroup from which they deduce the strong Feller property; while in [35] it is shown that the OU process Y has a càdlàg modification if and only if ∞ n=1 |β n | α < ∞.…”