On Harnack Inequality and Hölder Regularity for Isotropic Unimodal Lévy Processes

“…This is based on estimates obtained recently in [9,10,18,20] and will be studied in detail in [19]. Other extensions can be obtained by allowing the Lévy kernel to depend on x or restricting it to a domain, as described in the following two examples.…”

Section: Z) (D) Every Non-negative Function F Which Is a Harmonic Fumentioning

confidence: 99%

Martin Kernels for Markov Processes with Jumps

Kwaśnicki

Juszczyszyn

2017

Potential Anal

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We prove the existence of boundary limits of ratios of positive harmonic functions for a wide class of Markov processes with jumps and irregular (possibly disconnected) domains of harmonicity, in the context of general metric measure spaces. As a corollary, we prove the uniqueness of the Martin kernel at each boundary point, that is, we identify the Martin boundary with the topological boundary. We also prove a Martin representation theorem for harmonic functions. Examples covered by our results include: strictly stable Lévy processes in R d with positive continuous density of the Lévy measure; stable-like processes in R d and in domains; and stable-like subordinate diffusions in metric measure spaces.

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“…When A ≡ 0, the random process (X t ) t≥0 is said to be a purely jump process. Note that the properties (2.2) jointly imply that (X t ) t≥0 is a strong Feller process, or equivalently, its one-dimensional distributions are absolutely continuous with respect to Lebesgue measure, i.e., there exist measurable transition probability densities [23,Lem. 4]) that there exist C 1 , C 2 > 0, independent of the process (i.e., of A and ν), such that…”

Section: Jump-paring Class Of Lévy Processesmentioning

confidence: 99%

Fall-Off of Eigenfunctions for Non-Local Schrödinger Operators with Decaying Potentials

Kaleta

Lőrinczi

2016

Potential Anal

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We study the spatial decay of eigenfunctions of non-local Schrödinger operators whose kinetic terms are generators of symmetric jump-paring Lévy processes with Kato-class potentials decaying at infinity. This class of processes has the property that the intensity of single large jumps dominates the intensity of all multiple large jumps. We find that the decay rates of eigenfunctions depend on the process via specific preference rates in particular jump scenarios, and depend on the potential through the distance of the corresponding eigenvalue from the edge of the continuous spectrum. We prove that the conditions of the jump-paring class imply that for all eigenvalues the corresponding positive eigenfunctions decay at most as rapidly as the Lévy intensity. This condition is sharp in the sense that if the jump-paring property fails to hold, then eigenfunction decay becomes slower than the decay of the Lévy intensity. We furthermore prove that under reasonable conditions the Lévy intensity also governs the upper bounds of eigenfunctions, and ground states are comparable with it, i.e., two-sided bounds hold. As an interesting consequence, we identify a sharp regime change in the decay of eigenfunctions as the Lévy intensity is varied from subexponential to exponential order, and dependent on the location of the eigenvalue, in the sense that through the transition Lévy intensity-driven decay becomes slower than the rate of decay of the Lévy intensity. Our approach is based on path integration and probabilistic potential theory techniques, and all results are also illustrated by specific examples.

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On Harnack Inequality and Hölder Regularity for Isotropic Unimodal Lévy Processes

Cited by 83 publications

References 23 publications

α-stable random walk has massive thorns

α-stable random walk has massive thorns

Martin Kernels for Markov Processes with Jumps

Fall-Off of Eigenfunctions for Non-Local Schrödinger Operators with Decaying Potentials

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