Small-time sharp bounds for kernels of convolution semigroups

Kaleta, Kamil; Sztonyk, Paweł

doi:10.1007/s11854-017-0023-6

Cited by 39 publications

(74 citation statements)

References 51 publications

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“…For some classes of Lévy processes (e.g. relativistic stable and tempered stable) it can be shown that the density has exponential decay exp(− | |), see the exact heat kernel estimates obtained by Kaleta & Sztonyk [24,25].…”

Section: Exists Is Infinitely Often Differentiable and Satisfies Thementioning

confidence: 86%

“…For some classes of Lévy processes (e.g. relativistic stable and tempered stable) it can be shown that the density has exponential decay

exp (- m | x |)

, see the exact heat kernel estimates obtained by Kaleta & Sztonyk . (iii)Let ψ be a continuous negative definite function with Lévy triplet

(b, 0, ν)

. If ψ satisfies the sector condition, then it is possible to give sufficient conditions in terms of fractional moments of

{ν false|}_{B false(0, 1 false)}

and

{ν false|}_{B {false(0, 1 false)}^{c}}

which ensure that ψ satisfies the growth condition for real z , cf.…”

Section: Heat Kernel Estimates For Lévy Processesmentioning

confidence: 99%

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Transition probabilities of Lévy‐type processes: Parametrix construction

Kühn

2018

Mathematische Nachrichten

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We present an existence result for Lévy‐type processes which requires only weak regularity assumptions on the symbol q(x,ξ) with respect to the space variable x. Applications range from existence and uniqueness results for Lévy‐driven SDEs with Hölder continuous coefficients to existence results for stable‐like processes and Lévy‐type processes with symbols of variable order. Moreover, we obtain heat kernel estimates for a class of Lévy and Lévy‐type processes. The paper includes an extensive list of Lévy(‐type) processes satisfying the assumptions of our results.

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Section: Exists Is Infinitely Often Differentiable and Satisfies Thementioning

confidence: 86%

“…For some classes of Lévy processes (e.g. relativistic stable and tempered stable) it can be shown that the density has exponential decay

exp (- m | x |)

, see the exact heat kernel estimates obtained by Kaleta & Sztonyk . (iii)Let ψ be a continuous negative definite function with Lévy triplet

(b, 0, ν)

. If ψ satisfies the sector condition, then it is possible to give sufficient conditions in terms of fractional moments of

{ν false|}_{B false(0, 1 false)}

and

{ν false|}_{B {false(0, 1 false)}^{c}}

which ensure that ψ satisfies the growth condition for real z , cf.…”

Section: Heat Kernel Estimates For Lévy Processesmentioning

confidence: 99%

Transition probabilities of Lévy‐type processes: Parametrix construction

Kühn

2018

Mathematische Nachrichten

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“…In the proofs below we will often use the fact that (4.9) and (3.25) imply lim inf |ξ |→∞ ψ(ξ )/ log |ξ | > 0, which guarantees that assumption (A2) holds. Indeed, observe that by [36,Lem. 5 (a)], (2.4) and (4.8),…”

Section: Specific Casesmentioning

confidence: 98%

“…The convolution condition (A1.3) has been introduced in [33] and proved to be a strong tool in studying large-scale properties of jump Lévy processes. Recently, in [36] it was also used to characterize the short-time behaviour of heat kernels for a large class of convolution semigroups. It can be easily checked that (A1.3) in fact implies (A1.2), see [36, Lem.…”

Section: Definition 21 (Symmetric Jump-paring Lévy Processes)mentioning

confidence: 99%

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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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