  This article investigates the spectral structure of the evolution operators
associated with the statistical description of stochastic processes possessing
finite propagation velocity. Generalized Poisson-Kac processes and L\'evy walks
are explicitly considered as paradigmatic examples of regular and anomalous
dynamics. A generic spectral feature of these processes is the
lower-boundedness of the real part of the eigenvalue spectrum, corresponding to
an upper limit for the spectral dispersion curve, physically expressing the
relaxation rate of a disturbance as a function of the wave vector. We analyze
also Generalized Poisson-Kac processes possessing a continuum of stochastic
states parametrized with respect to the velocity. In this case, there exists a
critical value of the wavevector above which the point spectrum ceases to
exist, and the relaxation dynamics becomes controlled by the essential part of
the spectrum. This model can be extended to the quantum case and, in point of
fact, it represents a simple and highlighting example of a sub-quantum dynamics
with hidden variables.
