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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

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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.
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.

Resumen

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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