A priori Holder 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 $\LL$ on $\R^d$: $$\LL u(x) =\frac12 \sum_{i, j=1}^d \frac{\partial}{\partial x_i} (a_{ij}(x) \frac{\partial}{\partial x_j}) + \lim_{\eps \downarrow 0} \int_{\{y\in \R^d: |y-x|>\eps\}} (u(y)-u(x)) J(x, y) dy,$$ where $A(x)=(a_{ij}(x))_{1\leq i, j\leq d}$ is a measurable $d\times d$ matrix-valued function on $\R^d$ that is uniform elliptic and bounded and $J$ is a symmetric measurable non-trivial non-negative kernel on $\R^d\times \R^d$ satisfying certain conditions. Corresponding to $\LL$ is a symmetric strong Markov process $X$ on $\R^d$ that has both the diffusion component and pure jump component. We establish a priori H\"older estimate for bounded parabolic functions of $\LL$ and parabolic Harnack principle for positive parabolic functions of $\LL$. Moreover, two-sided sharp heat kernel estimates are derived for such operator $\LL$ 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.