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Often in applications such as rare events estimation or optimal control it isrequired that one calculates the principal eigen-function and eigen-value of anon-negative integral kernel. Except in the finite-dimensional case, usuallyneither the principal eigen-function nor the eigen-value can be computedexactly. In this paper, we develop numerical approximations for thesequantities. We show how a generic interacting particle algorithm can be used todeliver numerical approximations of the eigen-quantities and the associatedso-called "twisted" Markov kernel as well as how these approximations arerelevant to the aforementioned applications. In addition, we study a collectionof random integral operators underlying the algorithm, address some of theirmean and path-wise properties, and obtain $L_{r}$ error estimates. Finally,numerical examples are provided in the context of importance sampling forcomputing tail probabilities of Markov chains and computing value functions fora class of stochastic optimal control problems.

Calculating principal eigen-functions of non-negative integral kernels: particle approximations and applications