A priori Hölder estimate, parabolic Harnack principle and heat kernel estimates for diffusions with jumps

In this paper, we consider the following type of non-local (pseudo-differential) operators L on R^d : {align*} L u(x) =& 1/2 ∑i, j=1^d ∂/∂ x_i (aᵢⱼ(x) ∂ u(x)/∂ x_j) \\ &\ + limε ↓ 0 ∫_{\{y∈ R^d: |y-x|>ε\}} (u(y)-u(x)) J(x, y) dy, {align*} where A(x)=(aᵢⱼ(x))1≤ i,j≤ d is a measurable d× d matrix-valued function on R^d that is uniformly elliptic and bounded and J is a symmetric measurable non-trivial non-negative kernel on R^d × R^d satisfying certain conditions. Corresponding to L is a symmetric strong Markov process X on R^d that has both the diffusion component and pure jump component. We establish a priori Hölder estimate for bounded parabolic functions of L and parabolic Harnack principle for positive parabolic functions of L . Moreover, two-sided sharp heat kernel estimates are derived for such operator L and jump-diffusion X . In particular, our results apply to the mixture of symmetric diffusion of uniformly elliptic divergence form operator and mixed stable-like processes on R^d . To establish these results, we employ methods from both probability theory and analysis.