Uniqueness in Law of the stochastic convolution process driven by Lévy noise

We will give a proof of the following fact. If $\mathfrak{A}_1$ and $\mathfrak{A}_2$, $\tilde η_1$ and $\tilde η_2$, $ξ_1$ and $ξ_2$ 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 $L^p([0,T],{L}^{p}(Z,ν;E))\times \CM_I([0,T]\times Z)$ of the pairs $(ξ_1,η_1)$ and $(ξ_2,η_2)$ %, $i=1,2$, are equal, and $u_1$ and $u_2$ are the corresponding stochastic convolution processes, then the laws on $ (\DD([0,T];X)\cap L^p([0,T];B)) \times L^p([0,T],{L}^{p}(Z,ν;E))\times \CM_I([0,T]\times Z) $, where $B \subset E \subset X$, of the triples $(u_i,ξ_i,η_i)$, $i=1,2$, are equal as well. By $\DD([0,T];X)$ we denote the Skorokhod space of $X$-valued processes.

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