, Volume 52, Issue 7, pp 1423-1445
Date: 05 Jul 2009

Symmetric jump processes and their heat kernel estimates

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We survey the recent development of the DeGiorgi-Nash-Moser-Aronson type theory for a class of symmetric jump processes (or equivalently, a class of symmetric integro-differential operators). We focus on the sharp two-sided estimates for the transition density functions (or heat kernels) of the processes, a priori Hölder estimate and parabolic Harnack inequalities for their parabolic functions. In contrast to the second order elliptic differential operator case, the methods to establish these properties for symmetric integro-differential operators are mainly probabilistic.

This work was supported by National Science Foundation of USA (Grant No. DMS-0600206)