# Revista Matemática Iberoamericana

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**Volume 26, Issue 2, 2010, pp. 551–589**

**DOI: 10.4171/RMI/609**

Published online: 2010-08-31

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

Zhen-Qing Chen^{[1]}and Takashi Kumagai

^{[2]}(1) University of Washington, Seattle, United States

(2) Kyoto University, Japan

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 $A(x)=(a_{ij}(x))_{1\leq i,j\leq d}$ 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.

*Keywords: *Symmetric Markov process, pseudo-differential operator, diffusion process, jump process, Lévy system, hitting probability, parabolic function, a priori Hölder estimate, parabolic Harnack inequality, transition density, heat kernel estimates

Chen Zhen-Qing, Kumagai Takashi: A priori Hölder estimate, parabolic Harnack principle and heat kernel estimates for diffusions with jumps. *Rev. Mat. Iberoam.* 26 (2010), 551-589. doi: 10.4171/RMI/609