Pointwise eigenfunction estimates and intrinsic ultracontractivity-type properties of Feynman–Kac semigroups for a class of Lévy processes
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Kaleta, Kamil [1] ; Lőrinczi, József [2]
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[1] Wrocław University of Technology
Wrocław University of Technology
Breslavia, Polonia
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[2] Loughborough University
Loughborough University
Charnwood District, Reino Unido
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- Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 43, Nº. 3, 2015, págs. 1350-1398
- Idioma: inglés
- DOI: 10.1214/13-AOP897
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Resumen
We introduce a class of Lévy processes subject to specific regularity conditions, and consider their Feynman–Kac semigroups given under a Kato-class potential. Using new techniques, first we analyze the rate of decay of eigenfunctions at infinity. We prove bounds on λ-subaveraging functions, from which we derive two-sided sharp pointwise estimates on the ground state, and obtain upper bounds on all other eigenfunctions. Next, by using these results, we analyze intrinsic ultracontractivity and related properties of the semigroup refining them by the concept of ground state domination and asymptotic versions. We establish the relationships of these properties, derive sharp necessary and sufficient conditions for their validity in terms of the behavior of the Lévy density and the potential at infinity, define the concept of borderline potential for the asymptotic properties and give probabilistic and variational characterizations. These results are amply illustrated by key examples.
