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This research explores the conditions under which Markov operators satisfy Doeblin’s condition and their spectral properties on the unit circle.

June 13, 2026Open Access

Doeblin’s Condition, -Mixing and Spectra of Convolution Operators on the Circle

Key Points

This research explores the conditions under which Markov operators satisfy Doeblin’s condition and their spectral properties on the unit circle.
Analyzed Markov operators defined by convolution with a probability measure on the unit circle.
Demonstrated relationships between Doeblin’s condition and the non-singularity of powers of the probability measure.
Examined the spectra of Markov operators across different L_p spaces.
Proved that Doeblin’s condition holds if and only if some power of the probability measure is non-singular.
Provided an example where the Markov operator exhibits rho-mixing yet fails to satisfy Doeblin’s condition.
Studied spectral characteristics in various L_p spaces in both rho-mixing and non-rho-mixing scenarios.

Abstract

Abstract We study the asymptotic behavior of Markov operators P_ P μ defined by convolution with a probability measure μ on the unit circle T T. We prove that when μ is adapted, P_ P μ satisfies Doeblin’s condition if and only if some power ᵏ μ k is non-singular. We give an example of a symmetric probability measure μ on T T, such that the reversible stationary chain induced by P_ P μ is ρ -mixing, but P_ P μ does not satisfy Doeblin’s condition. We look at the spectra of P_ P μ in the different Lₚ L p spaces when P_ P μ is, or is not, ρ -mixing.

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