Molecular deuterium fluxes into the edge of deuterium-fuelled L-mode discharges are measured using passive visible spectroscopy of D(2) emission lines. Comparison with the atomic deuterium influx measured using D(alpha) emission suggests that a significant fraction of the plasma edge fuelling from the walls is in the form of D(2.) 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 D(2) excitation are measured, with ground state D(2) rotational population temperatures T(rot) up to 10000K and vibrational population temperatures T(vib) up to 30000 K. Comparisons between rotational population temperatures and the local electron density suggest that T(rot) can be used as a reasonably good indicator of electron density in the D(2) line emission region. In recombining, detached divertor operation, estimates of the enhanced volume recombination rate due to the presence of vibrationally-excited D(2) suggest that the effect of molecular-assisted volume recombination could be comparable in magnitude to that of normal D(+) volume recombination (EIR).