  In this paper, we construct operator fractional L\'evy motion (ofLm), a broad
class of non-Gaussian stochastic processes that are covariance operator
self-similar, have wide-sense stationary increments and display infinitely
divisible marginal distributions. The ofLm class generalizes the univariate
fractional L\'evy motion as well as the multivariate operator fractional
Brownian motion (ofBm). The ofLm class can be divided into two types, namely,
moving average (maofLm) and real harmonizable (rhofLm), both of which share the
covariance structure of ofBm under assumptions. We show that maofLm and rhofLm
admit stochastic integral representations in the time and Fourier domains, and
establish their distinct small- and large-scale limiting behavior. We
characterize time reversibility for ofLm through parametric conditions related
to its L\'evy measure, starting from a framework for the uniqueness of finite
second moment, multivariate stochastic integral representations. In particular,
we show that, under non-Gaussianity, the parametric conditions for time
reversibility are generally more restrictive than those for the Gaussian case
(ofBm).
