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September 2019 Intertwining, excursion theory and Krein theory of strings for non-self-adjoint Markov semigroups
Pierre Patie, Mladen Savov, Yixuan Zhao
Ann. Probab. 47(5): 3231-3277 (September 2019). DOI: 10.1214/19-AOP1338

Abstract

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.

Citation

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Pierre Patie. Mladen Savov. Yixuan Zhao. "Intertwining, excursion theory and Krein theory of strings for non-self-adjoint Markov semigroups." Ann. Probab. 47 (5) 3231 - 3277, September 2019. https://doi.org/10.1214/19-AOP1338

Information

Received: 1 August 2017; Revised: 1 December 2018; Published: September 2019
First available in Project Euclid: 22 October 2019

zbMATH: 07145316
MathSciNet: MR4021250
Digital Object Identifier: 10.1214/19-AOP1338

Subjects:
Primary: 37A30, 47D07
Secondary: 60G18

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.47 • No. 5 • September 2019
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