Revista Matemática Iberoamericana

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

Zhen-Qing Chen and Takashi Kumagai

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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.

Article information

Source
Rev. Mat. Iberoamericana, Volume 26, Number 2 (2010), 551-589.

Dates
First available in Project Euclid: 4 June 2010

Permanent link to this document
https://projecteuclid.org/euclid.rmi/1275671311

Mathematical Reviews number (MathSciNet)
MR2677007

Zentralblatt MATH identifier
1200.60065

Subjects
Primary: 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07] 47G30: Pseudodifferential operators [See also 35Sxx, 58Jxx] 60J45: Probabilistic potential theory [See also 31Cxx, 31D05]
Secondary: 31C05: Harmonic, subharmonic, superharmonic functions 31C25: Dirichlet spaces 60J75: Jump processes

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

Citation

Chen, Zhen-Qing; Kumagai, Takashi. A priori Hölder estimate, parabolic Harnack principle and heat kernel estimates for diffusions with jumps. Rev. Mat. Iberoamericana 26 (2010), no. 2, 551--589. https://projecteuclid.org/euclid.rmi/1275671311


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