A priori Hölder estimate, parabolic Harnack principle and heat kernel estimates for diffusions with jumps
Zhen-Qing Chen, Takashi Kumagai · Revista Matemática Iberoamericana · 2010
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.