Global heat kernel estimates for Δ+Δ α/2 in half-space-like domains

Zhen Qing Chen, Panki Kim, Renming Song

Research output: Contribution to journalArticlepeer-review

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 αΔ 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 αΔ α/2; a ∈ (0, 1]} in half-space-like C 1,1 domains in R d.

Original languageEnglish (US)
JournalElectronic Journal of Probability
Volume17
DOIs
StatePublished - 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

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