  We provide a mathematical study of the modified Diffusion Monte Carlo (DMC)
algorithm introduced in the companion article \cite{DMC}. DMC is a simulation
technique that uses branching particle systems to represent expectations
associated with Feynman-Kac formulae. We provide a detailed heuristic
explanation of why, in cases in which a stochastic integral appears in the
Feynman-Kac formula (e.g. in rare event simulation, continuous time filtering,
and other settings), the new algorithm is expected to converge in a suitable
sense to a limiting process as the time interval between branching steps goes
to 0. The situation studied here stands in stark contrast to the "na\"ive"
generalisation of the DMC algorithm which would lead to an exponential
explosion of the number of particles, thus precluding the existence of any
finite limiting object. Convergence is shown rigorously in the simplest
possible situation of a random walk, biased by a linear potential. The
resulting limiting object, which we call the "Brownian fan", is a very natural
new mathematical object of independent interest.
