Elliptic Harnack inequalities for symmetric non-local Dirichlet forms

Chen, Zhen-Qing; Kumagai, Takashi; Wang, Jian

doi:10.48550/arxiv.1703.09385

Cited by 4 publications

(11 citation statements)

References 25 publications

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“…As mentioned earlier, parabolic Harnack inequality is equivalent to the two-sided Aronson type heat kernel estimates for diffusion processes. In subsequent papers [CKW,CKW2], we study stability of parabolic Harnack inequality and elliptic Harnack inequality respectively for symmetric jump processes on metric measure spaces.…”

Section: Heat Kernelmentioning

confidence: 99%

Stability of heat kernel estimates for symmetric non-local Dirichlet forms

Chen¹,

Kumagai²,

Wang³

2016

Preprint

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In this paper, we consider symmetric jump processes of mixed-type on metric measure spaces under general volume doubling condition, and establish stability of two-sided heat kernel estimates and heat kernel upper bounds. We obtain their stable equivalent characterizations in terms of the jumping kernels, variants of cut-off Sobolev inequalities, and the Faber-Krahn inequalities. In particular, we establish stability of heat kernel estimates for α-stable-like processes even with α ≥ 2 when the underlying spaces have walk dimensions larger than 2, which has been one of the major open problems in this area.

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Section: Heat Kernelmentioning

confidence: 99%

Stability of heat kernel estimates for symmetric non-local Dirichlet forms

Chen¹,

Kumagai²,

Wang³

2016

Preprint

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“…(1.2) Theorem 1.2 was shown under the additional assumption that a kernel J(x, dy) exists and satisfies (1.2) in [CKW17, Proposition 3.3] (see also [BBK09,Proposition 4.7] for a similar result and proof). This assumption can be viewed as a weaker form of the existence of jump kernel and was assumed throughout [CKW17]. As a consequence of Theorem 1.2, we could remove the assumption (1.2) in the characterization of parabolic Harnack inequality in [CKW17].…”

Section: Framework and Resultsmentioning

confidence: 99%

“…This assumption can be viewed as a weaker form of the existence of jump kernel and was assumed throughout [CKW17]. As a consequence of Theorem 1.2, we could remove the assumption (1.2) in the characterization of parabolic Harnack inequality in [CKW17].…”

Section: Framework and Resultsmentioning

confidence: 99%

“…This characterization was extended and modified to many settings including diffusions on metric measure spaces [Stu], nearest neighbor walks on graphs [Del], and for anomalous space time scaling [BB,BBK06,GHL] by several authors. A similar characterization of the parabolic Harnack inequality for jump processes remained open until a recent breakthrough by Chen, Kumagai and Wang [CKW17].…”

Section: Introductionmentioning

confidence: 96%

“…In other words, we show that the jumping measure is absolutely continuous with respect to the product measure µ⊗µ, where µ is the symmetric (reference) measure for the jump process. As a consequence, we remove a technical hypothesis on the jumping measure assumed in [CKW17] for characterizing the parabolic Harnack inequality (see Remark 1.3).…”

Section: Introductionmentioning

confidence: 99%

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Parabolic Harnack inequality implies the existence of jump kernel

Liu

Murugan

2020

Preprint

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We prove that the parabolic Harnack inequality implies the existence of jump kernel for symmetric pure jump process. This allows us to remove a technical assumption on the jumping measure in the recent characterization of the parabolic Harnack inequality for pure jump processes by Chen, Kumagai and Wang. The key ingredients of our proof are the Lévy system formula and a near-diagonal heat kernel lower bound.

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Heat kernel estimates for anomalous heavy-tailed random walks

Murugan¹,

Saloff‐Coste²

2019

Ann. Inst. H. Poincaré Probab. Statist.

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Sub-Gaussian estimates for the natural random walk is typical of many regular fractal graphs. Subordination shows that there exist heavy tailed jump processes whose jump indices are greater than or equal to two. However, the existing machinery used to prove heat kernel bounds for such heavy tailed random walks fail in this case. In this work we extend Davies' perturbation method to obtain transition probability bounds for these anomalous heavy tailed random walks. We prove global upper and lower bounds on the transition probability density that are sharp up to constants. An important feature of our work is that the methods we develop are robust to small perturbations of the symmetric jump kernel.Résumé. Pour de nombreux graphes réguliers de type fractal, la marche aléatoire simple satisfait de estimation de type sous-Gaussiennes. La technique de la subordination montre alors qu'il existe the processus tête type stage dont l'indice des sauts est supérieur ouégale a 2. Pour de telles processus, les techniques usuelles pour les estimations loin de la diagonale ne fonctionnent pas. Nouś etendons la célèbre méthode de Davies dans le cas de ces processusà sauts "anormaux". Nous obtenons des bornes supérieures et infrieures précises sur le noyaux de transition pas des méthodes qui sont stables sous de petites perturbations des sauts.

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Elliptic Harnack inequalities for symmetric non-local Dirichlet forms

Cited by 4 publications

References 25 publications

Stability of heat kernel estimates for symmetric non-local Dirichlet forms

Stability of heat kernel estimates for symmetric non-local Dirichlet forms

Parabolic Harnack inequality implies the existence of jump kernel

Heat kernel estimates for anomalous heavy-tailed random walks

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