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 (pseudodifferential) operators £ on ℝ^d: Equation Presented where A(x) = (aij(x))i<i;j<d is a measurable dxd 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 x ℝ^d satisfying certain conditions. Corresponding to £ is a symmetric strong Markov process X on ℝ^d that has both the diffusion component and pure jump component. We establish a priori Holder estimate for bounded parabolic functions of C and parabolic Harnack principle for positive parabolic functions of C. Moreover, two-sided sharp heat kernel estimates are derived for such operator £ 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.