## Abstract

In this paper, we consider the following type of non-local (pseudodifferential) operators £ on ℝ^{d}: Equation Presented where A(x) = (aij(x))i<i;j<d is a measurable dxd matrix-valued function on ℝ^{d} that is uniformly elliptic and bounded and J is a symmetric measurable non-trivial non-negative kernel on ℝ^{d} x ℝ^{d} satisfying certain conditions. Corresponding to £ is a symmetric strong Markov process X on ℝ^{d} 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 ℝ^{d}. To establish these results, we employ methods from both probability theory and analysis.

Original language | English |
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Pages (from-to) | 551-589 |

Number of pages | 39 |

Journal | Revista Matematica Iberoamericana |

Volume | 26 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2010 |

Externally published | Yes |

## 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)