Molecular deuterium fluxes into the edge of deuterium-fuelled L-mode discharges are measured using passive visible spectroscopy of D2 emission lines. Comparison with the atomic deuterium influx measured using Dα emission suggests that a significant fraction of the plasma edge fuelling from the walls is in the form of D2. Molecular deuterium flux is observed in both the divertor and main chamber regions but is roughly a factor 100 smaller near the inner main chamber wall and roughly a factor 1000 smaller near the outer main chamber wall, when compared with the divertor region. Very high levels of molecular D2 excitation are measured, with ground state D2 rotational population temperatures Trot up to 10 000 K and vibrational population temperatures Tvib up to 30 000 K. Comparisons between rotational population temperatures and the local electron density suggest that Trot can be used as a reasonably good indicator of electron density in the D2 line emission region. In recombining, detached divertor operation, estimates of the enhanced volume recombination rate due to the presence of vibrationally-excited D2 suggest that the effect of molecular-assisted volume recombination could be comparable in magnitude to that of normal D+ volume recombination (EIR).