The radiation belts are thought to be produced through multiple processes. One of the original leading ideas was the concept of radial diffusion. Other ideas included energization due to low frequency waves. I have done work in this field, but only with a higher frequency electromagnetic wave called whistler mode waves. I will focus on these in my answer, but be aware that there are other mechanisms that can energize particles to MeV energies.
Though it may seem redundant to use mode and wave to describe the same thing, this results from the naming scheme set up originally for such phenomena. Early radio receivers were able to hear descending tones that sounded a lot like someone whistling, which led to the name whistlers. These were later found to be driven by electron beams produced during lightning strikes in thunderstorms in Earth's atmosphere. Now the term whistler is often synonymous with lightning whistler in the magnetospheric community.
With the advent of scientific spacecraft, two more types of whistler mode waves were discovered. These were called chorus and hiss (or plasmaspheric hiss). Their names imply what they sounded like when the original electromagnetic measurements were sonified. Chorus sound like a chorus of chirping birds and hiss sound like white noise, basically. The word plasmaspheric in front of hiss arose because it was discovered that the hissy signals seemed to be confined to the plasmasphere. Regardless, these two types of whistler mode waves have been studied since the 1960's and multiple theory and observation studies found them to be important in radiation belt dynamics.
More recently, large amplitude whistler mode waves were discovered in Earth's radiation belts. These were distinguished from chorus and hiss because their frequency did not drift in time like chorus, and they were coherent oscillations unlike hiss. Thus, the original papers specifically used the name whistler mode waves. The waves are so much larger than they were originally thought possible, that our entire understanding of this region had to be re-examined. Some simple test particle simulations showed that these very large amplitude waves could accelerate particles up to MeV energies in fractions of a second.
Later studies showed that these waves could produce a nonlinear phase trapping, thus quickly accelerating particles up to hundreds of keV very quickly. Over the years, ground stations would occassionally observe intense bursts of x-rays which were later determined to arise from bremstrahlung radiation from high energy incident electrons. These were called electron microbursts and were eventually associated with a specific type of whistler mode wave called chorus. Recent observations found a more definitive correlation between whistler mode waves and electron microbursts, supporting the discovery paper and subsequent study.
Shortly after, the properties of these very large amplitude whistler mode waves were categorized. This study found a relationship between the waves and injections of keV electrons from the geomagnetic tail. They also found that not all of these large amplitude whistler mode waves have constant frequencies with respect to time. The wave amplitudes were found to depend upon geomagnetic activity.
These results led to a monograph summarizing the results. The interesting part is that these high frequency ($\geq$100 Hz to $\geq$1000 Hz, depending on location in magnetosphere) waves can modify the outer radiation belts on time scales much shorter than the original estimates of days to weeks. In fact, more sophisticated studies than the original discovery paper found that such large amplitude waves could accelerate keV electrons to MeV energies in milliseconds.
The most recent results suggest that the large amplitude, coherent whistler mode waves are driven by the free energy introduced during substorm injections, supporting earlier suggestions.
So in regards to the Jovian system, the processes may be similar but the amount of free energy in the that magnetosphere is larger by orders of magnitude. Jupiter has a magnetic field ~14 times that of Earth and it spins on its axis every once every ~10 hours (even though its mean radius is >10 times that of Earth). The high rotation rate and large magnetic field provide very conducive environments for all the above processes to take place. Given the results from previous studies, it is not surprising that Jupiter has a very energetic set of radiation belts.
Fun Side Note
All NASA spacecraft that intend to use Jupiter for gravity assists need to undergo extra radiation hardening due to the harsh Jovian environments.