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June, 2010 A priori Hölder estimate, parabolic Harnack principle and heat kernel estimates for diffusions with jumps
Zhen-Qing Chen , Takashi Kumagai
Rev. Mat. Iberoamericana 26(2): 551-589 (June, 2010).

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

Citation

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Zhen-Qing Chen . Takashi Kumagai . "A priori Hölder estimate, parabolic Harnack principle and heat kernel estimates for diffusions with jumps." Rev. Mat. Iberoamericana 26 (2) 551 - 589, June, 2010.

Information

Published: June, 2010
First available in Project Euclid: 4 June 2010

zbMATH: 1200.60065
MathSciNet: MR2677007

Subjects:
Primary: 47G30, 60J35, 60J45
Secondary: 31C05, 31C25, 60J75

Rights: Copyright © 2010 Departamento de Matemáticas, Universidad Autónoma de Madrid

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Vol.26 • No. 2 • June, 2010
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