  We propose an adaptive importance sampling scheme for the simulation of rare
events when the underlying dynamics is given by a diffusion. The scheme is
based on a Gibbs variational principle that is used to determine the optimal
(i.e. zero-variance) change of measure and exploits the fact that the latter
can be rephrased as a stochastic optimal control problem. The control problem
can be solved by a stochastic approximation algorithm, using the Feynman-Kac
representation of the associated dynamic programming equations, and we discuss
numerical aspects for high-dimensional problems along with simple toy examples.
