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Sharp bounds on the density, Green function and jumping function of subordinate killed BM

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Abstract.

Subordination of a killed Brownian motion in a domain D⊂ℝd via an α/2-stable subordinator gives rise to a process Z t whose infinitesimal generator is −(−Δ| D )α/2, the fractional power of the negative Dirichlet Laplacian. In this paper we establish upper and lower estimates for the density, Green function and jumping function of Z t when D is either a bounded C 1,1 domain or an exterior C 1,1 domain. Our estimates are sharp in the sense that the upper and lower estimates differ only by a multiplicative constant.

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Author information

Correspondence to Renming Song.

Additional information

Mathematics Subject Classification (2000): Primary 60J45, Secondary 60J75, 31C25

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Song, R. Sharp bounds on the density, Green function and jumping function of subordinate killed BM. Probab. Theory Relat. Fields 128, 606–628 (2004). https://doi.org/10.1007/s00440-003-0316-9

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Keywords

  •  Killed Brownian motions
  • Stable processes
  • Subordination
  • Fractional Laplacian
  • Transition density
  • Green function
  • Jumping function