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.
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