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

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.

Published at http://dx.doi.org/10.1214/13-AOP897 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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• Boundary Harnack inequality for Markov processes with jumps
• Lieb-Thirring Bound for Schrödinger Operators with Bernstein Functions of the Laplacian
• Uniform Boundary Harnack Principle for Rotationally Symmetric Levy processes in General Open Sets
• Heat Trace of non-local operators
• Intrinsic ultracontractivity of nonsymmetric diffusions with measure-valued drifts and potentials
• A priori Holder estimate, parabolic Harnack principle and heat kernel estimates for diffusions with jumps

Cited by in corpus (15)

• Killing (absorption) versus survival in random motion
• Fall-off of eigenfunctions for non-local Schrödinger operators with decaying potentials
• Transition in the decay rates of stationary distributions of Lévy motion in an energy landscape
• Integrated density of states for Poisson-Schrödinger perturbations of subordinate Brownian motions on the Sierpiński gasket
• Multifractal properties of sample paths of ground state-transformed jump processes
• Kato classes for Lévy processes
• Quasi-ergodicity of compact strong Feller semigroups on $L^2$
• Intrinsic Ultracontractivity of Feynman-Kac Semigroups for Symmetric Jump Processes
• Spin-Boson Model through a Poisson-Driven Stochastic Process
• Typical long time behaviour of ground state-transformed jump processes
• Higher Order Eigenvalues for Non-Local Schrödinger Operators
• Intrinsic Ultracontractivity for General Lévy processes on Bounded Open Sets
• Intrinsic Ultracontractivity of Non-local Dirichlet forms on Unbounded Open Sets
• Periodic Solutions of Generalized Schrödinger Equations on Cayley Trees
• Intrinsic Contractivity of Feynman-Kac Semigroups for Symmetric Jump Processes with Infinite Range Jumps