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

By Zhen-Qing Chen, Takashi Kumagai. 2008

102 citations recorded by the source corpus.

In this paper, we consider the following type of non-local (pseudo-differential) operators $\LL $ on $\R^d$: $$ \LL u(x) =\frac12 \sum_{i, j=1}^d \frac{\partial}{\partial x_i} (a_{ij}(x) \frac{\partial}{\partial x_j}) + \lim_{\eps \downarrow 0} \int_{\{y\in \R^d: |y-x|>\eps\}} (u(y)-u(x)) J(x, y) dy, $$ where $A(x)=(a_{ij}(x))_{1\leq i, j\leq d}$ is a measurable $d\times d$ matrix-valued function on $\R^d$ that is uniform elliptic and bounded and $J$ is a symmetric measurable non-trivial non-negative kernel on $\R^d\times \R^d$ satisfying certain conditions. Corresponding to $\LL$ is a symmetric strong Markov process $X$ on $\R^d$ that has both the diffusion component and pure jump component. We establish a priori H\"older estimate for bounded parabolic functions of $\LL$ and parabolic Harnack principle for positive parabolic functions of $\LL$. Moreover, two-sided sharp heat kernel estimates are derived for such operator $\LL$ 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 $\R^d$. To establish these results, we employ methods from both probability theory and analysis.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/67532f30de487d7a4f62f0620e06986450e2a24c

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In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

On the conservativeness and the recurrence of symmetric jump-diffusions (2012, background): “Kumagai [3] for the Dirichlet form which has both the strongly-local and non-local terms.”
On Harnack Inequality and Hölder Regularity for Isotropic Unimodal Lévy Processes (2013, background): “• Since we do not exclude a case when a Gaussian part is non-zero we mention the paper [8], where diffusions with jumps were considered.”
Higher Hölder regularity for mixed local and nonlocal degenerate elliptic equations (2022, background): “6), BarlowBass-Chen-Kassmann [3], Chen-Kumagai [16] proved a Harnack inequality and local Hölder continuity.”
On the regularity theory for mixed local and nonlocal quasilinear elliptic equations (2021, background): “In contrast to the approach from probability and analysis introduced in [3,15], we study the problem (1.”
Boundary Harnack principle for Δ+Δ^{/2} (2012, background): “Proof. See Lemmas 2.2 and 2.3 in [16] or Lemmas 2.3 and 2.4 in [ 9 ] for a proof.”