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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.

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