Towards de Giorgi-Nash-Moser theory on non-linear non-local partial differential equations.

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We investigated recently developed methods on the de Giorgi-Nash-Moser theory and a priori estimates of caloric functions for non-local operators and jump-type stochastic processes. We gave some sufficient conditions for the boundary Harnack inequalities to hold for jump-type processes (non-local operators) on general metric measure spaces, and applied the results to various concrete examples. We also worked on heat kernel estimates for fractional time derivative heat equations.