Regularity of solutions to linear SPDEs driven by Lévy process
Seminar Room 1, Newton Institute
AbstractThe talk will be concerned with the following SPDE dX = AXdt + dL where A is the generator of a strongly continuous semigroup S on a Hilbert space H, and L is a Levy process taking values in a Hilbert space U into which H is embedded. If U = H, and S is a semigroup of contractions, then the existence of a cadlag version of X can be deduced from classical Kotelenez results. In many interesting cases U is strictly bigger than H. If the Levy measure $\mu$ of L is not supported on H, than it is easy to show that X cannot have locally bounded, and hence c`adl`ag (or even weakly cadlag) trajectories in H. During the talk a natural case of L leaving in living in $U \not= H$, but with jumps of the size from H will be considered. The existence of a weakly cadlag version will be shown. Examples of equations with and without cadlag versions of solutions will be given. The talk will be based on the following papers: S. Peszat, Cadlag version of an infinite-dimmensional Ornstein.Uhlenbeck process driven by Levy noise, preprint. Z. BrzeLzniak, B. Goldys, P. Imkeller, S. Peszat, E. Priola, J. Zabczyk, Time irregularity of generalized Ornstein.Uhlenbeck processes, submitted.
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