  Airy beams are known for displaying shape invariance and self-acceleration
along the transverse direction while they propagate forwards. Although these
properties could be associated with the beam coherence, it has been revealed
that they also manifest in the case of partially coherent Airy-type. Here,
these properties are further investigated by introducing and analyzing a class
of partially coherent Airy beams under both infinite and finite energy
conditions. The key element within the present approach is the so-called
cross-spectral density, which enables a direct connection with the quantum
density matrix, making the analysis exportable to the quantum realm to study
the dynamics of Airy wave packets acted by both incoherence and decoherence. As
it is shown, in the case of infinite energy beams both properties are preserved
even under the circumstance of total incoherence provided the underlying
structure of the beam remains equal to that of an Airy beam. In the case of
finite energy beams, a situation closer to a realistic scenario, as
experimental beams cannot have an infinite extension, it is shown that a
propagation range along which both properties are preserved can be warranted.
This is controlled by a critical distance, which depends on the spread range
determined by the parameters ruling the extension of random field spatial
fluctuations. Such a distance is determined by defining a position-dependent
parameter that quantifies the degree of overlapping between the propagated beam
and the input one displaced by an amount equivalent to the propagation
distance.
