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Heat kernel estimates for the Dirichlet fractional Laplacian

Zhen-Qing Chen, Panki Kim, Renming Song (2010)

Journal of the European Mathematical Society

We consider the fractional Laplacian - ( - Δ ) α / 2 on an open subset in d with zero exterior condition. We establish sharp two-sided estimates for the heat kernel of such a Dirichlet fractional Laplacian in C 1 , 1 open sets. This heat kernel is also the transition density of a rotationally symmetric α -stable process killed upon leaving a C 1 , 1 open set. Our results are the first sharp twosided estimates for the Dirichlet heat kernel of a non-local operator on open sets.

Heat kernel of fractional Laplacian in cones

Krzysztof Bogdan, Tomasz Grzywny (2010)

Colloquium Mathematicae

We give sharp estimates for the transition density of the isotropic stable Lévy process killed when leaving a right circular cone.

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