Elliptic Harnack inequalities for symmetric non-local Dirichlet forms

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

doi:10.1016/j.matpur.2017.10.011

Cited by 18 publications

(19 citation statements)

References 57 publications

(148 reference statements)

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“…In particular, the process X satisfies condition (7) in [23,Theorem 1.20]. Now, using [23,Theorem 1.20] again we obtain E Φ and UHK(Φ). This completes the proof.…”

Section: From the Above Two Inequalities We Conclude The Lemma ✷mentioning

confidence: 81%

“…Since the jump kernel of X and Y are comparable by Lemma 4.2, the conditions PI(Φ) and CSJ(Φ) also hold for X. In particular, the process X satisfies condition (7) in [23,Theorem 1.20]. Now, using [23,Theorem 1.20] again we obtain E Φ and UHK(Φ).…”

Section: From the Above Two Inequalities We Conclude The Lemma ✷mentioning

confidence: 83%

“…Thus, by [22, Theorem 1.15], we see that CSJ(Φ) defined in [22] holds. Thus, using J ψ,≤ , PI(Φ), CSJ(Φ), (UJS) and (2.6), we have (7) in [23,Theorem 1.20].…”

Section: Proofs Of Theorems 213 and 214mentioning

confidence: 99%

“…Note that from Lemma 4.2 and Lemma 4.5, the condition (4) in [23,Theorem 1.20] holds for the process Y . In particular, using [23,Theorem 1.20], the conditions CSJ(Φ) and PI(Φ) holds for the process Y . Since the jump kernel of X and Y are comparable by Lemma 4.2, the conditions PI(Φ) and CSJ(Φ) also hold for X.…”

Section: From the Above Two Inequalities We Conclude The Lemma ✷mentioning

confidence: 99%

“…Recently, heat kernel estimates for Markov processes with jumps have been extensively studied due to their importance in theory and applications (see [8,10,3,12,14,16,19,20,21,24,22,23,25,26,29,30,32,33,35,36,38,39,40,41,44,45,46,47,49,51,52,55,56] and references therein). In [21], the authors obtained heat kernel estimates for pure jump symmetric Markov processes on metric measure space (M, d, µ), where M is a locally compact separable metric measure space with metric d and a positive Radon measure µ satisfying volume doubling property.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Heat kernel estimates for symmetric jump processes with mixed polynomial growths

Bae¹,

Kang²,

Kim³

et al. 2019

Ann. Probab.

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In this paper, we consider a symmetric pure jump Markov process X on a general metric measure space that satisfies volume doubling conditions. We study estimates of the transition density p(t, x, y) of X and their stabilities when the jumping kernel for X has general mixed polynomial growths. Unlike [22], in our setting, the rate function which gives growth of jumps of X may not be comparable to the scale function which provides the borderline for p(t, x, y) to have either near-diagonal estimates or off-diagonal estimates. Under the assumption that the lower scaling index of scale function is strictly bigger than 1, we establish stabilities of heat kernel estimates. If underlying metric measure space admits a conservative diffusion process which has a transition density satisfying a general sub-Gaussian bounds, we obtain heat kernel estimates which generalize [3, Theorems 1.2 and 1.4]. In this case, scale function is explicitly given by the rate function and the function F related to walk dimension of underlying space. As an application, we prove that the finite moment condition in terms of F on such symmetric Markov process is equivalent to a generalized version of Khintchine-type law of iterated logarithm at infinity.

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“…In particular, the process X satisfies condition (7) in [23,Theorem 1.20]. Now, using [23,Theorem 1.20] again we obtain E Φ and UHK(Φ). This completes the proof.…”

Section: From the Above Two Inequalities We Conclude The Lemma ✷mentioning

confidence: 81%

Section: From the Above Two Inequalities We Conclude The Lemma ✷mentioning

confidence: 83%

“…Thus, by [22, Theorem 1.15], we see that CSJ(Φ) defined in [22] holds. Thus, using J ψ,≤ , PI(Φ), CSJ(Φ), (UJS) and (2.6), we have (7) in [23,Theorem 1.20].…”

Section: Proofs Of Theorems 213 and 214mentioning

confidence: 99%

Section: From the Above Two Inequalities We Conclude The Lemma ✷mentioning

confidence: 99%