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

Abstract

In this paper, we consider the following type of non-local (pseudo-differential) operators $\mathcal{L}$ on ${\mathbb{R}}^d$: \begin{align*} \mathcal{L} u(x) = \frac{1}{2} \sum_{i, j=1}^d \frac{\partial}{\partial x_i} \Big(a_{ij}(x) \frac{\partial u(x)}{\partial x_j}\Big) \\+ \lim_{\varepsilon \downarrow 0} \int_{\{y\in \mathbb{R}^d: |y-x|>\varepsilon\}} (u(y)-u(x)) J(x, y) dy, \end{align*} where $A(x)=(a_{ij}(x){)}_{1\le i,j\le d}$ is a measurable $d\times d$ matrix-valued function on ${\mathbb{R}}^d$ that is uniformly elliptic and bounded and $J$ is a symmetric measurable non-trivial non-negative kernel on ${\mathbb{R}}^d\times {\mathbb{R}}^d$ satisfying certain conditions. Corresponding to $\mathcal{L}$ is a symmetric strong Markov process $X$ on ${\mathbb{R}}^d$ that has both the diffusion component and pure jump component. We establish a priori Hölder estimate for bounded parabolic functions of $\mathcal{L}$ and parabolic Harnack principle for positive parabolic functions of $\mathcal{L}$. Moreover, two-sided sharp heat kernel estimates are derived for such operator $\mathcal{L}$ 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 ${\mathbb{R}}^d$. To establish these results, we employ methods from both probability theory and analysis.