Open Access
June 2009 Gaussian perturbations of circle maps: A spectral approach
John Mayberry
Ann. Appl. Probab. 19(3): 1143-1171 (June 2009). DOI: 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 λ-bifurcation. Asymptotic expressions for the corresponding eigenfunctions and eigenmeasures are also derived and are related to Hermite functions.


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John Mayberry. "Gaussian perturbations of circle maps: A spectral approach." Ann. Appl. Probab. 19 (3) 1143 - 1171, June 2009.


Published: June 2009
First available in Project Euclid: 15 June 2009

zbMATH: 1189.60130
MathSciNet: MR2537202
Digital Object Identifier: 10.1214/08-AAP573

Primary: 37H20 , 60J05
Secondary: 47A55

Keywords: Eigenvalues , integrate-and-fire models , Markov chains , pseudospectra , Random perturbations , stochastic bifurcations , transition operators

Rights: Copyright © 2009 Institute of Mathematical Statistics

Vol.19 • No. 3 • June 2009
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