Time regularity of Lévy-type evolution in Hilbert spaces and of some $α$-stable processes

In this paper we consider the existence of weakly càdlàg versions of a solution to a linear equation in a Hilbert space $H$, driven by a Levy process taking values in a Hilbert space $U$. In particular we are interested in diagonal type processes, where process on coordinates are functionals of independent $α$ stable symmetric process. We give the if and only if characterization in this case. We apply the same techniques to obtain a sufficient condition for existence of a càdlàg versions of stable processes described as integrals of deterministic functions with respect to symmetric $α$-stable random measures with $α\in[1,2)$.

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