Calculating Principal Eigen-Functions of Non-Negative Integral Kernels: Particle Approximations and Applications

Abstract

Often in applications such as rare events estimation or optimal control it is required that one calculates the principal eigenfunction and eigenvalue of a nonnegative integral kernel. Except in the finite-dimensional case, usually neither the principal eigenfunction nor the eigenvalue can be computed exactly. In this paper, we develop numerical approximations for these quantities. We show how a generic interacting particle algorithm can be used to deliver numerical approximations of the eigenquantities and the associated so-called “twisted” Markov kernel as well as how these approximations are relevant to the aforementioned applications. In addition, we study a collection of random integral operators underlying the algorithm, address some of their mean and pathwise properties, and obtain error estimates. Finally, numerical examples are provided in the context of importance sampling for computing tail probabilities of Markov chains and computing value functions for a class of stochastic optimal control problems.