  We study the acceleration, transport, and emission of particles in
relativistic jets. Localized stochastic particle acceleration, spatial
diffusion, and synchrotron as well as synchrotron self-Compton emission are
considered in a leptonic model. To account for inhomogeneity, we use a 2D
axi-symmetric cylindrical geometry for both relativistic electrons and magnetic
field. In this first phase of our work, we focus on steady-state spectra that
develop from a time-dependent model. We demonstrate that small isolated
acceleration region in a much larger emission volume are sufficient to
accelerate particles to high energy. Diffusive escape from these small regions
provides a natural explanation for the spectral form of the jet emission. The
location of the acceleration regions within the jet is found to affect the
cooling break of the spectrum in this diffusive model. Diffusion-caused
energy-dependent inhomogeneity in the jets predicts that the SSC spectrum is
harder than the synchrotron spectrum. There can also be a spectral hardening
towards the high-energy section of the synchrotron spectrum, if particle escape
is relatively slow. These two spectral hardening effects indicate that the jet
inhomogeneity might be a natural explanation for the unexpected hard
{\gamma}-ray spectra observed in some blazars.
