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Diffusion processes with small noise conditioned to reach a target set areconsidered. The AMS algorithm is a Monte Carlo method that is used to samplesuch rare events by iteratively simulating clones of the process and selectingtrajectories that have reached the highest value of a so-called importancefunction. In this paper, the large sample size relative variance of the AMSsmall probability estimator is considered. The main result is a largedeviations logarithmic equivalent of the latter in the small noise asymptotics,which is rigorously derived. It is given as a maximisation problem explicit interms of the quasi-potential cost function associated with the underlying smallnoise large deviations. Necessary and sufficient geometric conditions ensuringthe vanishing of the obtained quantity ('weak' asymptotic efficiency) areprovided. Interpretations and practical consequences are discussed.

Fluctuations of Rare Event Simulation with Monte Carlo Splitting in the Small Noise Asymptotics