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Uniqueness in law of the stochastic convolution process driven by Lévy noise. (English) Zbl 1284.60125
Summary: We give a proof of the following fact. If \(\mathfrak{A}_1\) and \(\mathfrak{A}_2\), \(\tilde \eta_1\) and \(\tilde \eta_2\), \(\xi_1\) and \(\xi_2\) are two examples of filtered probability spaces, time homogeneous compensated Poisson random measures, and progressively measurable Banach space-valued processes such that the laws on \(L^p([0,T],{L}^{p}(Z,\nu ;E))\times \mathcal{M}_I([0,T]\times Z)\) of the pairs \((\xi_1,\eta_1)\) and \((\xi_2,\eta_2)\), are equal, and \(u_1\) and \(u_2\) are the corresponding stochastic convolution processes, then the laws on \( (\mathbb{D}([0,T];X)\cap L^p([0,T];B)) \times L^p([0,T],{L}^{p}(Z,\nu ;E))\times \mathcal{M}_I([0,T]\times Z) \), where \(B \subset E \subset X\), of the triples \((u_i,\xi_i,\eta_i)\), \(i=1,2\), are equal as well. By \(\mathbb{D}([0,T];X)\) we denote the Skorokhod space of \(X\)-valued processes.

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60G57 Random measures
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