A priori hölder estimate, parabolic harnack principle and heat kernel estimates for diffusions with jumps

Zhen Qing Chen*, Takashi Kumagai

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

34 Citations (Scopus)

Abstract

In this paper, we consider the following type of non-local (pseudodifferential) operators £ on ℝd: Equation Presented where A(x) = (aij(x))i<i;j<d is a measurable dxd matrix-valued function on ℝd that is uniformly elliptic and bounded and J is a symmetric measurable non-trivial non-negative kernel on ℝd x ℝd satisfying certain conditions. Corresponding to £ is a symmetric strong Markov process X on ℝd that has both the diffusion component and pure jump component. We establish a priori Holder estimate for bounded parabolic functions of C and parabolic Harnack principle for positive parabolic functions of C. Moreover, two-sided sharp heat kernel estimates are derived for such operator £ 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 ℝd. To establish these results, we employ methods from both probability theory and analysis.

Original languageEnglish
Pages (from-to)551-589
Number of pages39
JournalRevista Matematica Iberoamericana
Volume26
Issue number2
DOIs
Publication statusPublished - 2010
Externally publishedYes

Keywords

  • A priori holder estimate
  • Diffusion process
  • Heat kernel estimates
  • Hitting probability
  • Jump process
  • Levy system
  • Parabolic function
  • Parabolic harnack inequality
  • Pseudo-differential operator
  • Symmetric markov process
  • Transition density

ASJC Scopus subject areas

  • Mathematics(all)

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