  In this note we develop a prelimit analysis of performance measures for
importance sampling schemes related to small noise diffusion processes. In
importance sampling the performance of any change of measure is characterized
by its second moment. For a given change of measure, we characterize the second
moment of the corresponding estimator as the solution to a PDE, which we
analyze via a full asymptotic expansion with respect to the size of the noise
and obtain a precise statement on its accuracy. The main correction term to the
decay rate of the second moment solves a transport equation that can be solved
explicitly. The asymptotic expansion that we obtain identifies the source of
possible poor performance of nevertheless asymptotically optimal importance
sampling schemes and allows for more accurate comparison among competing
importance sampling schemes.
