  This article is concerned with the mathematical analysis of a family of
adaptive importance sampling algorithms applied to diffusion processes. These
methods, referred to as Adaptive Biasing Potential methods, are designed to
efficiently sample the invariant distribution of the diffusion process, thanks
to the approximation of the associated free energy function (relative to a
reaction coordinate). The bias which is introduced in the dynamics is computed
adaptively; it depends on the past of the trajectory of the process through
some time-averages.
  We give a detailed and general construction of such methods. We prove the
consistency of the approach (almost sure convergence of well-chosen weighted
empirical probability distribution). We justify the efficiency thanks to
several qualitative and quantitative additional arguments. To prove these
results , we revisit and extend tools from stochastic approximation applied to
self-interacting diffusions, in an original context.
