Heat kernel estimates for the Dirichlet fractional Laplacian

Zhen Qing Chen, Panki Kim, Renming Song

Research output: Contribution to journalArticlepeer-review


We consider the fractional Laplacian - (-Δ) α/2 on an open subset in ℝ 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.

Original languageEnglish (US)
Pages (from-to)1307-1327
Number of pages21
JournalJournal of the European Mathematical Society
Issue number5
StatePublished - 2010


  • Boundary harnack inequality
  • Exit time
  • Fractional Laplacian
  • Green function
  • Heat kernel
  • Intrinsic ultracontractivity
  • Lévy system
  • Parabolic harnack inequality
  • Symmetric α-stable process
  • Transition density

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics


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