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On linear evolution equations for a class of cylindrical Lévy noises. (English) Zbl 1278.60095
Da Prato, Giuseppe (ed.) et al., Stochastic partial differential equations and applications. Papers based on the presentations at the 8th meeting, Levico Terme, Italy, January 2008. Caserta: Dipartimento di Matematica, Seconda Università di Napoli (ISBN 978-88-548-4391-2/hbk). Quaderni di Matematica 25, 223-242 (2010).
The authors consider Hilbert-valued linear stochastic differential equations driven by cylindrical Levy noise, i.e., equations of the form \[ dX_t = AX_t dt+dZ_t, X_0=x\in H, t\geq 0, \] where \(A\) is the generator of a \(C_0\)-semigroup on the Hilbert space \(H\) and \(Z\) is cylindrical Levy process (taking values in a Hilbert space \(U\) possibly larger than \(H\)), which is assumed to have no Gaussian part. The authors provide sufficient and necessary conditions under which solutions to the above equations take values in \(H\) - using standard assumptions on the operator \(A\). As an example the stochastic heat equation with Dirichlet boundary conditions is considered. Furthermore the Markov property and the irreducibility of the solution are established.
For the entire collection see [Zbl 1250.60003].

MSC:
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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