Does current in a superconducting circuit ever decay? Once a current has been started in a superconducting loop, say by induction, will it truly stay on forever or is it intrinsically doomed to decay, albeit slowly?
For example, can radiative losses eventually damp the persistent current?
 A: A persistent current in a ring is thermodynamically a meta-stable state. It is separated by a barrier from states with smaller number of flux quanta. This means that at any non-zero temperature the persistent current state will eventually decay (by tunneling or thermal activation) to a state with no current, and the energy dissipated as heat. However, this requires macroscopic fluctuations (involving the motion of many electrons), and it is therefore extremely unlikely. 
Note that a DC current is not subject to radiation damping. 
A: Below the critical temperature a superconductor has no loss, or ohmic, resistance.  With no resistance, there would be no heat produced to radiate away the current, and an induced current would persist forever.
Practically speaking, however, any impurities in the superconductor would create loss resistance, which would cause heat to radiate and the current to be attenuated, albeit slowly.
In addition to heat radiation due to a conductor's ohmic resistance, one also can find radiation resistance.  For example, AC current in an antenna radiates energy in the form of electromagnetic waves due to acceleration of charges in the conductor.  Radiation resistance depends on the shape and length of a conductor as compared to the wavelength of the current flowing through it.
It may be possible to design a superconducting loop that minimizes radiation resistance. 
A: Supercurrents are a flow of the Bose-Einstein-Condensate (BEC), where each boson is an electron pair. BEC-bosons have a minimum and quantized kinetic energy and, thus, cannot emit their energy by arbitrarily small portions. So below BEC temperature the pairs don't lose any energy and the supercurrent flows forever.
