  Cylindrical probability measures are finitely additive measures on Banach
spaces that have sigma-additive projections to Euclidean spaces of all
dimensions. They are naturally associated to notions of weak (cylindrical)
random variable and hence weak (cylindrical) stochastic processes. In this
paper we focus on cylindrical Levy processes. These have (weak) Levy-Ito
decompositions and an associated Levy-Khintchine formula. If the process is
weakly square integrable, its covariance operator can be used to construct a
reproducing kernel Hilbert space in which the process has a decomposition as an
infinite series built from a sequence of uncorrelated bona fide one-dimensional
Levy processes. This series is used to define cylindrical stochastic integrals
from which cylindrical Ornstein-Uhlenbeck processes may be constructed as
unique solutions of the associated Cauchy problem. We demonstrate that such
processes are cylindrical Markov processes and study their (cylindrical)
invariant measures.
