The stochastic Cauchy problem driven by a cylindrical Lévy process. (English) Zbl 1445.60045
Summary: In this work, we derive sufficient and necessary conditions for the existence of a weak and mild solution of an abstract stochastic Cauchy problem driven by an arbitrary cylindrical Lévy process. Our approach requires to establish a stochastic Fubini result for stochastic integrals with respect to cylindrical Lévy processes. This approach enables us to conclude that the solution process has almost surely scalarly square integrable paths. Further properties of the solution such as the Markov property and stochastic continuity are derived.
MSC:
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
| 60G51 | Processes with independent increments; Lévy processes |
| 60G20 | Generalized stochastic processes |
| 60H05 | Stochastic integrals |
Keywords:
cylindrical Lévy process; Cauchy problem; stochastic Fubini theorem; cylindrical infinitely divisibleReferences:
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