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 estimates on the heat kernel.

中文翻译：

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抛物线型Harnack不等式意味着存在跳跃核

我们证明抛物型Harnack不等式暗示对称纯跳跃过程存在跳跃核。这使我们可以消除对Chen，Kumagai和Wang的纯跳跃过程的抛物型Harnack不等式的最近刻画中关于跳跃测度的技术假设。我们证明的关键要素是Lévy系统公式和对热核的估计。