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

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dc.creatorChen, Zhen-Qing
dc.creatorKumagai, Takashi
dc.date2008-08-29
dc.date.accessioned2026-07-07T09:59:24Z
dc.date.available2026-07-07T09:59:24Z
dc.descriptionIn 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 Hölder 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.
dc.description32 pages
dc.identifierhttps://arxiv.org/abs/0808.4010
dc.identifierhttp://arxiv.org/abs/0808.4010
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/168009
dc.subjectProbability
dc.subjectAnalysis of PDEs
dc.subject60J45, 60J35 (Primary); 31C05, 31C25, 60J75 (Secondary)
dc.titleA priori Holder estimate, parabolic Harnack principle and heat kernel estimates for diffusions with jumps
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