We consider a d-dimensional pure jump Markov process M with a jumping kernel comparable to a d-dimensional anisotropic L'evy process L, where the coordinates of L are independent 1-dimensional L´evy processes. We obtain the sharp two-sided bounds of the fundamental solution (heat kernel) for the non-local operators corresponding to the pure jump Markov processes M. This talk is based on two projects: a joint work with Moritz Kassmann and Takashi Kumagai, and a joint work with Lidan Wang. The first is a study of the Markov process where each coordinate is comparable to the α-stable process, and the second is a study of the general Markov process in which the L´evy kernel has the weakly scaling condition.

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