  Considering the issue of estimating small probabilities p, ie. measuring a
rare domain F = {x | g(x) > q} with respect to the distribution of a random
vector X, Multilevel Splitting strategies (also called Subset Simulation) aim
at writing F as an intersection of less rare events (nested subsets) such that
their measures are conditionally easily computable. However the definition of
an appropriate sequence of nested subsets remains an open issue.
  We introduce here a new approach to Multilevel Splitting methods in terms of
a move of particles in the input space. This allows us to derive two main
results: (1) the number of samples required to get a realisation of X in F is
drastically reduced, following a Poisson law with parameter log 1/p (to be
compared with 1/p for naive Monte-Carlo); and (2) we get a parallel optimal
Multilevel Splitting algorithm where there is indeed no subset to define any
more.
  We also apply result (1) in quantile estimation producing a new parallel
algorithm and derive a new strategy for the construction of first Design Of
Experiments in meta-model based algorithms.
