On non-local ergodic Jacobi semigroups: spectral theory, convergence-to-equilibrium and contractivity

Cheridito, Patrick; Patie, Pierre; Srapionyan, Anna; Vaidyanathan, Aditya

Journal de l’École polytechnique — Mathématiques, Tome 8 (2021), pp. 331-378.

Dans cet article, nous introduisons et étudions des opérateurs de Jacobi non locaux, qui généralisent les opérateurs de Jacobi classiques (locaux). Nous montrons que ces opérateurs s’étendent aux générateurs de semi-groupes de Markov ergodiques avec des mesures de probabilité invariantes uniques et étudions leurs propriétés spectrales et de convergence. En particulier, nous dérivons un développement en série du semi-groupe en termes de polynômes explicitement définis, qui généralisent les polynômes orthogonaux de Jacobi classiques. De plus, nous donnons une caractérisation complète du spectre du générateur et du semi-groupe non auto-adjoint. Nous montrons que la convergence de la variance du semi-groupe est hypocoercive avec des constantes explicites, ce qui fournit une généralisation naturelle de l’estimation donnée par le trou spectral. Après un temps de préchauffage aléatoire, le semi-groupe décroît également de manière exponentielle en entropie et est à la fois hypercontractif et ultracontractif. Nos preuves s’articulent autour du développement d’identités de commutation, appelées relations d’entrelacement, entre opérateurs et semi-groupes de Jacobi locaux et non locaux, les objets locaux servant de points de référence pour le transfert de propriétés du cas local au cas non local.

In this paper, we introduce and study non-local Jacobi operators, which generalize the classical (local) Jacobi operators. We show that these operators extend to generators of ergodic Markov semigroups with unique invariant probability measures and study their spectral and convergence properties. In particular, we derive a series expansion of the semigroup in terms of explicitly defined polynomials, which generalize the classical Jacobi orthogonal polynomials. In addition, we give a complete characterization of the spectrum of the non-self-adjoint generator and semigroup. We show that the variance decay of the semigroup is hypocoercive with explicit constants, which provides a natural generalization of the spectral gap estimate. After a random warm-up time, the semigroup also decays exponentially in entropy and is both hypercontractive and ultracontractive. Our proofs hinge on the development of commutation identities, known as intertwining relations, between local and non-local Jacobi operators and semigroups, with the local objects serving as reference points for transferring properties from the local to the non-local case.

Classification : 37A30, 47D06, 47G20, 60J75
Keywords: Markov semigroups, spectral theory, non-self-adjoint operators, convergence to equilibrium, hypercontractivity, ultracontractivity, heat kernel estimates
Mot clés : Semi-groupes de Markov, théorie spectrale, opérateurs non auto-adjoints, convergence vers l’équilibre, hypercontractivité, ultracontractivité, estimations du noyau de la chaleur

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