  A significant fraction of the mature FGK stars have cool dusty disks at least
an orders of magnitudes brighter than the solar system's outer zodiacal light.
Since such dusts must be continually replenished, they are generally assumed to
be the collisional fragments of residual planetesimals analogous to the Kuiper
Belt objects. At least 10% of solar type stars also bear gas giant planets. The
fraction of stars with known gas giants or detectable debris disks (or both)
appears to increase with the stellar mass. Here, we examine the dynamical
evolution of systems of long-period gas giant planets and residual
planetesimals as their host stars evolve off the main sequence, lose mass, and
form planetary nebula around remnant white dwarf cores. The orbits of distant
gas giant planets and super-km-size planetesimals expand adiabatically. During
the most intense AGB mass loss phase, sub-meter-size particles migrate toward
their host stars due to the strong hydrodynamical drag by the intense stellar
wind. Along their migration paths, gas giant planets capture and sweep up
sub-km-size planetesimals onto their mean-motion resonances. These
planetesimals also acquire modest eccentricities which are determined by the
mass of the perturbing planets, the rate and speed of stellar mass loss. The
swept-up planetesimals undergo disruptive collisions which lead to the
production of grains with an extended size range. The radiation drag on these
particles is ineffective against the planets' resonant barrier and they form
30-to-150-AU-sizes rings which can effective reprocess the stellar irradiation
in the form of FIR continuum. We identify the recently discovered dust ring
around the white dwarf WD 2226-210 at the center of the Helix nebula as a
prototype of such disks and suggest such rings may be common.
