## Abstract

Suppose that d ≥ 1 and α ∈ (0, 2). In this paper, we establish by using probabilistic methods sharp two-sided pointwise estimates for the Dirichlet heat kernels of {Δ + a ^{α}Δ ^{α/2}; a ∈ (0, 1]} on half-space-like C ^{1,1} domains for all time t > 0. The large time estimates for half-space-like domains are very different from those for bounded domains. Our estimates are uniform in a ∈ (0, 1] in the sense that the constants in the estimates are independent of a ∈ (0, 1]. Thus they yield the Dirichlet heat kernel estimates for Brownian motion in half-space-like domains by taking a → 0. Integrating the heat kernel estimates with respect to the time variable t, we obtain uniform sharp two-sided estimates for the Green functions of {Δ+a ^{α}Δ ^{α/2}; a ∈ (0, 1]} in half-space-like C ^{1,1} domains in R ^{d}.

Original language | English (US) |
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Journal | Electronic Journal of Probability |

Volume | 17 |

DOIs | |

State | Published - 2012 |

## Keywords

- Brownian motion
- Exit time
- Fractional laplacian
- Green function
- Harmonic function
- Heat kernel
- Laplacian
- Lévy system
- Symmetric α-stable process
- Transition density

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty