  Nowadays many tools, e.g. fluctuation relations, are available to
characterize the statistical properties of non-equilibrium systems. However,
most of these tools rely on the assumption that the driving noise is normally
distributed. Here we consider a class of Markov processes described by Langevin
equations driven by a mixture of Gaussian and Poissonian noises, focusing on
their non-equilibrium properties. In particular, we prove that detailed balance
does not hold even when correlation functions are symmetric under time
reversal. In such cases, a breakdown of the time reversal symmetry can be
highlighted by considering higher order correlation functions. Furthermore, the
entropy production may be different from zero even for vanishing currents. We
provide analytical expressions for the average entropy production rate in
several cases. We also introduce a scale dependent estimate for entropy
production, suitable for inference from experimental signals. The empirical
entropy production allows us to discuss the role of spatial and temporal
resolutions in characterizing non-equilibrium features. Finally, we revisit the
Brownian gyrator introducing an additional Poissonian noise showing that it
behaves as a two dimensional linear ratchet. It has also the property that when
Onsager relations are satisfied its entropy production is positive although it
is minimal. We conclude discussing estimates of entropy production for
partially accessible systems, comparing our results with the lower bound
provided by the thermodynamic uncertainty relations.
