## Abstract

We consider a family of pseudo differential operators {Δ+a^{α}Δ^{α/2}; a ∈ [0, 1]} on ℝ^{d} that evolves continuously from Δ to Δ + Δ^{α/2}, where d ≥ 1 and α ∈ (0, 2). It gives rise to a family of Lévy processes {X^{a}, a ∈ [0, 1]}, where X^{a} is the sum of a Brownian motion and an independent symmetric α-stable process with weight a. Using a recently obtained uniform boundary Harnack principle with explicit decay rate, we establish sharp bounds for the Green function of the process X^{a} killed upon exiting a bounded C^{1,1} open set D ⊂ ℝ^{d}. Our estimates are uniform in a ∈ (0, 1] and taking a→0 recovers the Green function estimates for Brownian motion in D. As a consequence of the Green function estimates for X^{a} in D, we identify both the Martin boundary and the minimal Martin boundary of D with respect to X^{a} with its Euclidean boundary. Finally, sharp Green function estimates are derived for certain Lévy processes which can be obtained as perturbations of X^{a}.

Original language | English |
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Pages (from-to) | 981-1024 |

Number of pages | 44 |

Journal | Illinois Journal of Mathematics |

Volume | 54 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2010 |

Externally published | Yes |

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