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

### Zhen-Qing Chen

University of Washington, Seattle, United States### Takashi Kumagai

Kyoto University, Japan

## Abstract

In this paper, we consider the following type of non-local (pseudo-differential) operators $L$ on $R_{d}$: \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 $A(x)=(a_{ij}(x))_{1≤i,j≤d}$ 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.

## Cite this article

Zhen-Qing Chen, Takashi Kumagai, A priori Hölder estimate, parabolic Harnack principle and heat kernel estimates for diffusions with jumps. Rev. Mat. Iberoam. 26 (2010), no. 2, pp. 551–589

DOI 10.4171/RMI/609