The Annals of Probability

Pointwise eigenfunction estimates and intrinsic ultracontractivity-type properties of Feynman–Kac semigroups for a class of Lévy processes

Kamil Kaleta and József Lőrinczi

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Abstract

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 $\lambda$-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.

Article information

Source
Ann. Probab., Volume 43, Number 3 (2015), 1350-1398.

Dates
First available in Project Euclid: 5 May 2015

Permanent link to this document
https://projecteuclid.org/euclid.aop/1430830284

Digital Object Identifier
doi:10.1214/13-AOP897

Mathematical Reviews number (MathSciNet)
MR3342665

Zentralblatt MATH identifier
1321.47098

Subjects
Primary: 47D08: Schrödinger and Feynman-Kac semigroups 60G51: Processes with independent increments; Lévy processes
Secondary: 47D03: Groups and semigroups of linear operators {For nonlinear operators, see 47H20; see also 20M20} 47G20: Integro-differential operators [See also 34K30, 35R09, 35R10, 45Jxx, 45Kxx]

Keywords
Symmetric Lévy process subordinate Brownian motion Feynman–Kac semigroup nonlocal operator intrinsic ultracontractivity entropy ground state domination decay of eigenfunctions $\lambda$-subaveraging function

Citation

Kaleta, Kamil; Lőrinczi, József. Pointwise eigenfunction estimates and intrinsic ultracontractivity-type properties of Feynman–Kac semigroups for a class of Lévy processes. Ann. Probab. 43 (2015), no. 3, 1350--1398. doi:10.1214/13-AOP897. https://projecteuclid.org/euclid.aop/1430830284


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