Jupiter-sized planet orbiting a white dwarf?

The orbiting planet has a short orbital period and might be subject to tidal dissipation and orbital shrinkage. However, tidal dissipation (i.e. the raising of tides by the planet on the white dwarf and consequent frictional/viscous loss of orbital energy) in a white dwarf is about 12 orders of magnitude less efficient than for a main sequence star, and similar, well-observed, close-in exoplanets around normal stars show little sign of orbital shrinkage (with one exception). The conclusion is therefore that this planet should be stable on timescales of billions of years.

The planet in question is a Jupiter-sized object (the mass is still uncertain, but likely in the planetary regime and not a brown dwarf), with an orbital period of just 34 hours.

The cool white dwarf is older than about 5 billion years. During its earlier red giant phase it would have engulfed and destroyed any planet orbiting with a period less than about a year, so the idea is that this planet migrated inwards after the star became a white dwarf. The mechanism by which it does this is that a planet with a wide orbit (orbital period > 1 year) avoids engulfment but has its orbital eccentricity excited by interaction with anothe planet or with a third body in the system. The eccentricity brings it close to the white dwarf at perihelion and tidal dissipation then causes the orbit to circularise at roughly the perihelion distance.

From the point of view of stability there is not really much difference between an exoplanet orbiting a normal star every 34 hours and a planet orbiting a white dwarf (with similar mass). There should be a tidal dissipation effect whereby if the rotation period of the star is different from the orbital period, then tides raised by the planet on the star will cause dissipation and shrinkage of the orbit. This effect will only be important for very close-in exoplanets. As far as I know, there is empirical evidence for this happening only in the case of the extreme example of WASP-12b with an orbital period of 26 hours. The orbit of this planet appears to be shrinking on a timescale of millions of years (Patra et al. 2017).

However, this effect should be much smaller for a white dwarf central star. The tidal dissipation factor is expected to be much lower for a compact, dense white dwarf - e.g. the rate of orbital shrinkage is proportional to the Love number, which controls tidal dissipation and the stellar radius to the power of 5. The Love number is $k_2 sim 0.01$ for a white dwarf, compared with $k_2sim 0.6$ for a main sequence star (e.g. Prodan & Murray 2012) and the radius of the star will be 100 times smaller than for a main sequence star, and so I expect that the orbit of this object will be stable on very long timescales from that point of view.

A bit more detail

From Patra et al. (2017) we can write an equation for the tidal shrinkage in the orbit as $$ frac{dP}{dt} propto left( frac{M_p}{M_*}right) left(frac{R_*}{a}right)^5 k_2,$$ where $M_p$ is the mass and radius of the planet, $M_*$ and $R_*$ are the mass and radius of the central star and $a$ is the orbital separation. Using Kepler's third law $a propto M_*^{1/3}P^{2/3}$, we can write $$frac{dP}{dt} propto M_p M_*^{-8/3} R_*^{5} P^{-10/3} k_2$$ If we compare WASP-12b with the planet orbiting the white dwarf, then we can (roughly) assume that the planetery masses, the stellar masses and the orbital periods are quite similar. However, the value of $k_2$ is $sim 100$ times smaller for the white dwarf and the white dwarf radius is $sim 100$ times smaller than WASP-12. The rate of change of orbital period is therefore 12 orders of magnitude smaller.