JournalsrmiVol. 26, No. 2pp. 551–589

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

In this paper, we consider the following type of non-local (pseudo-differential) operators L\mathcal{L} on Rd\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)=(aij(x))1i,jdA(x)=(a_{ij}(x))_{1\leq i,j\leq d} is a measurable d×dd\times d matrix-valued function on Rd\mathbb{R}^d that is uniformly elliptic and bounded and JJ is a symmetric measurable non-trivial non-negative kernel on Rd×Rd\mathbb{R}^d \times \mathbb{R}^d satisfying certain conditions. Corresponding to L\mathcal{L} is a symmetric strong Markov process XX on Rd\mathbb{R}^d that has both the diffusion component and pure jump component. We establish a priori Hölder estimate for bounded parabolic functions of L\mathcal{L} and parabolic Harnack principle for positive parabolic functions of L\mathcal{L}. Moreover, two-sided sharp heat kernel estimates are derived for such operator L\mathcal{L} and jump-diffusion XX. In particular, our results apply to the mixture of symmetric diffusion of uniformly elliptic divergence form operator and mixed stable-like processes on Rd\mathbb{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