Gaussian perturbations of circle maps: A spectral approach

Details Gaussian perturbations of circle maps: A spectral approach Gaussian perturbations of circle maps: A spectral approach

2009-08-07

arxiv.org/abs/0908.1058v1

Annals of Applied Probability 2009, Vol. 19, No. 3, 1143-1171

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/08-AAP573 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Inst

Scientific paper

10.1214/08-AAP573

In this work, we examine spectral properties of Markov transition operators corresponding to Gaussian perturbations of discrete time dynamical systems on the circle. We develop a method for calculating asymptotic expressions for eigenvalues (in the zero noise limit) and show that changes to the number or period of stable orbits for the deterministic system correspond to changes in the number of limiting modulus 1 eigenvalues of the transition operator for the perturbed process. We call this phenomenon a $\lambda$-bifurcation. Asymptotic expressions for the corresponding eigenfunctions and eigenmeasures are also derived and are related to Hermite functions.

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