Document Type

Article

Publication Title

Annals of Applied Probability

Department

Mathematics

ISSN

1050-5164

Volume

19

Issue

3

DOI

10.1214/08-AAP573

First Page

1143

Last Page

1171

Publication Date

6-1-2009

Abstract

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 λ-bifurcation. Asymptotic expressions for the corresponding eigenfunctions and eigenmeasures are also derived and are related to Hermite functions. © Institute of Mathematical Statistics, 2009.

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