  Direct imaging searches have revealed many very low-mass objects, including a
small number of planetary mass objects, as wide-orbit companions to young
stars. The formation mechanism of these objects remains uncertain. In this
paper we present the predictions of the disc fragmentation model regarding the
properties of the discs around such low-mass objects. We find that the discs
around objects that have formed by fragmentation in discs hosted by Sun-like
stars (referred to as 'parent' discs and 'parent' stars) are more massive than
expected from the ${M}_{\rm disc}-M_*$ relation (which is derived for stars
with masses $M_*>0.2 {\rm M}_{\odot}$). Accordingly, the accretion rates onto
these objects are also higher than expected from the $\dot{M}_*-M_*$ relation.
Moreover there is no significant correlation between the mass of the brown
dwarf or planet with the mass of its disc nor with the accretion rate from the
disc onto it. The discs around objects that form by disc fragmentation have
larger than expected masses as they accrete gas from the disc of their parent
star during the first few kyr after they form. The amount of gas that they
accrete and therefore their mass depend on how they move in their parent disc
and how they interact with it. Observations of disc masses and accretion rates
onto very low-mass objects are consistent with the predictions of the disc
fragmentation model. Future observations (e.g. by ALMA) of disc masses and
accretion rates onto substellar objects that have even lower masses (young
planets and young, low-mass brown dwarfs), where the scaling relations
predicted by the disc fragmentation model diverge significantly from the
corresponding relations established for higher-mass stars, will test the
predictions of this model.
