  We complement and extend our work on fluctuation relations arising in
nonequilibrium systems in steady states driven by L\'evy noise [Phys. Rev. E
76, 020101(R) (2006)]. As a concrete example, we consider a particle subjected
to a drag force and a L\'evy white noise with tail index $\mu\in (0,2]$, and
calculate the probability distribution of the work done on the particle by the
drag force, as well as the probability distribution of the work dissipated by
the dragged particle in a nonequilibrium steady state. For $0<\mu<2$, both
distributions satisfy what we call an anomalous fluctuation relation,
characterized by positive and negative fluctuations that asymptotically have
the same probability. For $\mu=2$, by contrast, the work and dissipated work
distributions satisfy the known conventional and extended fluctuation
relations, respectively, which are both characterized by positive fluctuations
that are exponentially more probable than negative fluctuations. The difference
between these different fluctuation relations is discussed in the context of
large deviation theory. Experiments that could probe or reveal anomalous
fluctuation relations are also discussed.
