The Annals of Probability

Intertwining, excursion theory and Krein theory of strings for non-self-adjoint Markov semigroups

Pierre Patie, Mladen Savov, and Yixuan Zhao

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In this paper, we start by showing that the intertwining relationship between two minimal Markov semigroups acting on Hilbert spaces implies that any recurrent extensions, in the sense of Itô, of these semigroups satisfy the same intertwining identity. Under mild additional assumptions on the intertwining operator, we prove that the converse also holds. This connection, which relies on the representation of excursion quantities as developed by Fitzsimmons and Getoor (Illinois J. Math. 50 (2006) 413–437), enables us to give an interesting probabilistic interpretation of intertwining relationships between Markov semigroups via excursion theory: two such recurrent extensions that intertwine share, under an appropriate normalization, the same local time at the boundary point. Moreover, in the case when one of the (non-self-adjoint) semigroup intertwines with the one of a quasi-diffusion, we obtain an extension of Krein’s theory of strings by showing that its densely defined spectral measure is absolutely continuous with respect to the measure appearing in the Stieltjes representation of the Laplace exponent of the inverse local time. Finally, we illustrate our results with the class of positive self-similar Markov semigroups and also the reflected generalized Laguerre semigroups. For the latter, we obtain their spectral decomposition and provide, under some conditions, an explicit hypocoercivity $L^{2}$-rate of convergence to equilibrium which is expressed as the spectral gap perturbed by the spectral projection norms.

Article information

Ann. Probab., Volume 47, Number 5 (2019), 3231-3277.

Received: August 2017
Revised: December 2018
First available in Project Euclid: 22 October 2019

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Mathematical Reviews number (MathSciNet)

Primary: 37A30: Ergodic theorems, spectral theory, Markov operators {For operator ergodic theory, see mainly 47A35} 47D07: Markov semigroups and applications to diffusion processes {For Markov processes, see 60Jxx}
Secondary: 60G18: Self-similar processes

Excursion theory intertwining Krein theory Markov semigroups self-similar spectral theory


Patie, Pierre; Savov, Mladen; Zhao, Yixuan. Intertwining, excursion theory and Krein theory of strings for non-self-adjoint Markov semigroups. Ann. Probab. 47 (2019), no. 5, 3231--3277. doi:10.1214/19-AOP1338.

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