Gaussian perturbations of circle maps: A spectral approach
John Mayberry
Source: Ann. Appl. Probab.
Volume 19, Number 3
(2009), 1143-1171.
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.
Primary Subjects: 60J05, 37H20
Secondary Subjects: 47A55
Keywords: Random perturbations; Markov chains; transition operators; stochastic bifurcations; integrate-and-fire models; eigenvalues; pseudospectra
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Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aoap/1245071022
Digital Object Identifier: doi:10.1214/08-AAP573
Zentralblatt MATH identifier:
05580236
Mathematical Reviews number (MathSciNet):
MR2537202
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