  Diffusive shock acceleration (DSA) at relativistic shocks is widely thought
to be an important acceleration mechanism in various astrophysical jet sources,
including radio-loud active galactic nuclei such as blazars. Such acceleration
can produce the non-thermal particles that emit the broadband continuum
radiation that is detected from extragalactic jets. An important recent
development for blazar science is the ability of Fermi-LAT spectroscopy to pin
down the shape of the distribution of the underlying non-thermal particle
population. This paper highlights how multi-wavelength spectra spanning optical
to X-ray to gamma-ray bands can be used to probe diffusive acceleration in
relativistic, oblique, magnetohydrodynamic (MHD) shocks in blazar jets.
Diagnostics on the MHD turbulence near such shocks are obtained using thermal
and non-thermal particle distributions resulting from detailed Monte Carlo
simulations of DSA. These probes are afforded by the characteristic property
that the synchrotron $\nu F_{\nu}$ peak energy does not appear in the gamma-ray
band above 100 MeV. We investigate self-consistently the radiative synchrotron
and inverse Compton signatures of the simulated particle distributions.
Important constraints on the diffusive mean free paths of electrons, and the
level of electromagnetic field turbulence are identified for three different
case study blazars, Mrk 501, BL Lacertae and AO 0235+164. The X-ray excess of
AO 0235+164 in a flare state can be modelled as the signature of bulk Compton
scattering of external radiation fields, thereby tightly constraining the
energy-dependence of the diffusion coefficient for electrons. The concomitant
interpretations that turbulence levels decline with remoteness from jet shocks,
and the probable significant role for non-gyroresonant diffusion, are posited.
