Relation between Diffusion in Momentum and Decoherence I have noticed in several sources (for example, eq. (3.151) of Decoherence and the Appearance of a Classical World in Quantum Theory and eq. (5.40) of Decoherence, einselection, and the quantum origins of the classical) that a diffusion in momentum of the Wigner function corresponds to decoherence. Why does diffusion in momentum imply decoherence?
I also remember reading that the disappearance of interference terms is indicative of decoherence for the Wigner function. If this is so, how does the disappearance of interference terms relate to a diffusion in momentum?
 A: It is easiest to start with the latter part of your question. Indeed, in the two classic references you are citing, it is emphasized repeatedly that the hallmark of QM is interference, "waves", so oscillatory phenomena of the Wigner Function, underlying the negative values of this quasiprobability. When a quantum system interacts with the environment, it is getting "observed" and increasingly with time decoheres to resemble "classical systems" which lack such interference. 
For the sake of specificity, you might consider the interference of two coherently superposed Gaussian wavepackets going through two slits spaced 2a apart, producing the standard Wigner Function of Fig 1 in Thomas L. Curtright, David B. Fairlie, & Cosmas K. Zachos, A Concise Treatise on Quantum Mechanics in Phase Space, World Scientific, 2014. In this case, observe cos(2 pa) oscillations in the momentum direction---not the x direction.

To simulate these random external "observations" in numerical experiments, both your references introduce, by hand, prototype diffusion terms, as in Zurek's (5.40), that is they augment the proper Moyal evolution equation with a diffusion damper term. 
If we ignored the QM evolution, so let the Diffusive term do the bulk of the driving (a driftless Fokker-Planck equation), we'd get a plain diffusion equation for the WF, (3.151) of your first ref, Joos et al.,
$$
\partial_t W =D ~\partial_{p}^2 W,
$$
whose propagator yields the Weierstrass transform, a  convolution with a Gaussian, so a low-pass filter damping oscillations, Joos' (3.152)
$$
W(x,p,t)=\frac{1}{4\pi D t }\int dk W(x,k,0) e^{-(k-p)^2/4Dt}.
$$
The Weierstrass transform of cos( 2 ap) is $e^{-4a^2Dt} \cos 2ap$ so, schematically, you can see the oscillatory terms in such expressions damp out with time and suppress the quantum dips below zero (the tell-tale quantum terms!), and thus appear increasingly classical: QM coherence is gone. In our particular example, the coherence of the two wavepackets is lost across the two slits, and the picture would be just two phase-space Gaussians with the middle oscillation mound filtered out! Note the net area of the mound was essentially zero, so probability has been preserved: The Weierstrass transform of a normalized distribution is normalized.
Electrical engineers actually play these games in time-frequency domain signal processing, considering the wave beats as some type of undesirable "interference". The formal procedure involved here is related to the Husimi representation, but this goes far off the subject...
Edit elicited by comment: In phase space QM there is no deep conceptual difference between p, x, or any combination thereof. The examples your references provide and I illustrate deal with p-oscillations generated by spacelike coherers, but they could have been chosen differently to produce x-oscillations instead. If your references featured x-oscillations, the filtering-by-hand insert term should have been chosen in the x-variable. Diffusion, however, works through the second derivative, as illustrated. For phase-space-rotated cat states, see, e.g., Braverman. 
