  In this work, we derive sufficient and necessary conditions for the existence
of a weak and mild solution of an abstract stochastic Cauchy problem driven by
an arbitrary cylindrical Levy process. Our approach requires to establish a
stochastic Fubini result for stochastic integrals with respect to cylindrical
Levy processes. This approach enables us to conclude that the solution process
has almost surely scalarly square integrable paths. Further properties of the
solution such as the Markov property and stochastic continuity are derived.
