  We consider particle filters with weakly informative observations (or
`potentials') relative to the latent state dynamics. The particular focus of
this work is on particle filters to approximate time-discretisations of
continuous-time Feynman--Kac path integral models -- a scenario that naturally
arises when addressing filtering and smoothing problems in continuous time --
but our findings are indicative about weakly informative settings beyond this
context too. We study the performance of different resampling schemes, such as
systematic resampling, SSP (Srinivasan sampling process) and stratified
resampling, as the time-discretisation becomes finer and also identify their
continuous-time limit, which is expressed as a suitably defined `infinitesimal
generator.' By contrasting these generators, we find that (certain
modifications of) systematic and SSP resampling `dominate' stratified and
independent `killing' resampling in terms of their limiting overall resampling
rate. The reduced intensity of resampling manifests itself in lower variance in
our numerical experiment. This efficiency result, through an ordering of the
resampling rate, is new to the literature. The second major contribution of
this work concerns the analysis of the limiting behaviour of the entire
population of particles of the particle filter as the time discretisation
becomes finer. We provide the first proof, under general conditions, that the
particle approximation of the discretised continuous-time Feynman--Kac path
integral models converges to a (uniformly weighted) continuous-time particle
system.
