Density and tails of unimodal convolution semigroups

Bogdan, Krzysztof; Grzywny, Tomasz; Ryznar, Michał

doi:10.1016/j.jfa.2014.01.007

Cited by 143 publications

(237 citation statements)

References 29 publications

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“…For any such ν there exists a unique pure-jump isotropic unimodal Lévy process X (see [3], [67]). The characteristic exponent Φ of X takes the form…”

Section: Appendix -Unimodal Lévy Processesmentioning

confidence: 99%

Heat kernels of non-symmetric Lévy-type operators

Grzywny

Szczypkowski

2019

Journal of Differential Equations

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“…For any such ν there exists a unique pure-jump isotropic unimodal Lévy process X (see [3], [67]). The characteristic exponent Φ of X takes the form…”

Section: Appendix -Unimodal Lévy Processesmentioning

confidence: 99%

Heat kernels of non-symmetric Lévy-type operators

Grzywny

Szczypkowski

2019

Journal of Differential Equations

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“…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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“…See [35,Definition 6.1] and [27,Theorem 13.3.5] for details. Moreover, we also have (1.7) when (1.6) holds for any 0 < r ≤ R < ∞ (See [3,Corollary 22]).…”

Section: Introductionmentioning

confidence: 93%

Boundary regularity for nonlocal operators with kernels of variable orders

Kim

Lee

et al. 2019

Journal of Functional Analysis

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We study the boundary regularity of solutions of the Dirichlet problem for the nonlocal operator with a kernel of variable orders. Since the order of differentiability of the kernel is not represented by a single number, we consider the generalized Hölder space. We prove that there exists a unique viscosity solution of Lu = f in D, u = 0 in R n \ D, where D is a bounded C 1,1 open set, and that the solution u satisfies u ∈ C V (D) and u/V (d D ) ∈ C α (D) with the uniform estimates, where V is the renewal function and d D (x) = dist(x, ∂D).

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Density and tails of unimodal convolution semigroups

Abstract: We give sharp bounds for the isotropic unimodal probability convolution semigroups when their Lévy-Khintchine exponent has Matuszewska indices strictly between 0 and 2.

Cited by 143 publications

References 29 publications

Heat kernels of non-symmetric Lévy-type operators

Heat kernels of non-symmetric Lévy-type operators

Martin Kernels for Markov Processes with Jumps

Boundary regularity for nonlocal operators with kernels of variable orders

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