  The origin of the extended X-ray emission in the large-scale jets of active
galactic nuclei (AGNs) poses challenges to conventional models of acceleration
and emission. Although the electron synchrotron radiation is considered the
most feasible radiation mechanism, the formation of the continuous large-scale
X-ray structure remains an open issue. As astrophysical jets are expected to
exhibit some turbulence and shearing motion, we here investigate the potential
of shearing flows to facilitate an extended acceleration of particles and
evaluate its impact on the resultant particle distribution. Our treatment
incorporates systematic shear and stochastic second-order Fermi effects. We
show that for typical parameters applicable to large-scale AGN jets, stochastic
second-order Fermi acceleration, which always accompanies shear particle
acceleration, can play an important role in facilitating the whole process of
particle energization. We study the time-dependent evolution of the resultant
particle distribution in the presence of second-order Fermi acceleration, shear
acceleration, and synchrotron losses using a simple Fokker--Planck approach and
provide illustrations for the possible emergence of a complex (multicomponent)
particle energy distribution with different spectral branches. We present
examples for typical parameters applicable to large-scale AGN jets, indicating
the relevance of the underlying processes for understanding the extended X-ray
emission and the origin of ultrahigh-energy cosmic rays.
