Time regularity of L\'{e}vy-type evolution in Hilbert spaces and of some $\alpha$-stable processes

TL;DR

This paper investigates the conditions under which solutions to certain Levy-driven linear equations in Hilbert spaces have right-continuous paths with left limits, focusing on diagonal processes and $eta$-stable processes.

Contribution

It provides a characterization for the existence of c extquoteright{}adl extquoteright{} versions of Levy-driven Hilbert space processes and extends to stable processes as integrals of deterministic functions.

Findings

Characterization of when solutions have c extquoteright{}adl extquoteright{} versions.

Sufficient conditions for c extquoteright{}adl extquoteright{} versions of stable processes.

Application to diagonal type processes with independent $eta$-stable components.

Abstract

In this paper we consider the existence of weakly c\`adl\`ag 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 $\alpha$ 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 c\`adl\`ag versions of stable processes described as integrals of deterministic functions with respect to symmetric $\alpha$-stable random measures with $\alpha\in[1,2)$.