We consider the problem of simulation-based estimation of
performance measures for a Markov chain conditioned on a rare
event. The conditional law depends on the solution of a
multiplicative Poisson equation. An adaptive scheme for learning
the latter is proposed and analyzed. An example motivated by a
well known problem in communication networks is given.
Applications of the basic scheme to other related domains such
as importance sampling for rare event simulation and the
solution of a class of eigenvalue problems are also sketched.