  The lifetime of a planetary disc which orbits a white dwarf represents a
crucial input parameter into evolutionary models of that system. Here we apply
a purely analytical formalism to estimate lifetimes of the debris phase of
these discs, before they are ground down into dust or are subject to
sublimation from the white dwarf. We compute maximum lifetimes for three
different types of white dwarf discs, formed from (i) radiative YORP breakup of
exo-asteroids along the giant branch phases at 2-100 au, (ii) radiation-less
spin-up disruption of these minor planets at $\sim 1.5-4.5R_{\odot}$, and (iii)
tidal disruption of minor or major planets within about $1.3R_{\odot}$. We
display these maximum lifetimes as a function of disc mass and extent,
constituent planetesimal properties, and representative orbital excitations of
eccentricity and inclination. We find that YORP discs with masses up to
$10^{24}$ kg live long enough to provide a reservoir of surviving cm-sized
pebbles and m- to km-sized boulders that can be perturbed intact to white
dwarfs with cooling ages of up to 10 Gyr. Debris discs formed from the spin or
tidal disruption of these minor planets or major planets can survive in a
steady state for up to respectively 1 Myr or 0.01 Myr, although most tidal
discs would leave a steady state within about 1 yr. Our results illustrate that
dust-less planetesimal transit detections are plausible, and would provide
particularly robust evolutionary constraints. Our formalism can easily be
adapted to individual systems and future discoveries.
