  In this paper, we consider an importance sampling problem for a certain
rare-event simulations involving the behavior of a diffusion process pertaining
to a chain of distributed systems with random perturbations. We also assume
that the distributed system formed by $n$-subsystems -- in which a small random
perturbation enters in the first subsystem and then subsequently transmitted to
the other subsystems -- satisfies an appropriate H\"{o}rmander condition. Here
we provide an efficient importance sampling estimator, with an exponential
variance decay rate, for the asymptotics of the probabilities of the rare
events involving such a diffusion process that also ensures a minimum relative
estimation error in the small noise limit. The framework for such an analysis
basically relies on the connection between the probability theory of large
deviations and the values functions for a family of stochastic control problems
associated with the underlying distributed system, where such a connection
provides a computational paradigm -- based on an exponentially-tilted biasing
distribution -- for constructing efficient importance sampling estimators for
the rare-event simulation. Moreover, as a by-product, the framework also allows
us to derive a family of Hamilton-Jacobi-Bellman for which we also provide a
solvability condition for the corresponding optimal control problem.
