On Feller's boundary problem for Markov processes in weak duality

Zhen Qing Chen*, Masatoshi Fukushima

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

7 Citations (Scopus)
Plum Print visual indicator of research metrics
  • Citations
    • Citation Indexes: 7
  • Captures
    • Readers: 3
see details

Abstract

We give an affirmative answer to Feller's boundary problem going back to 1957 by obtaining a resolvent characterization for the duality preserving extensions of a pair of standard Markov processes in weak duality (minimal processes) to the boundary consisting of countably many points. Our resolvent characterization involves the resolvents for the minimal processes, the Feller measures that are intrinsic to the minimal processes as well as the restrictions to the boundary of the jumping and killing measures of the extension processes. Conversely, given killing rates on the boundary, we construct the corresponding duality preserving extensions of the minimal processes that admit no jumps between the boundary points and have the prescribed killing rate at the boundary, by repeatedly doing one-point extension one at a time using Itô's Poisson point processes of excursions.

Original languageEnglish
Pages (from-to)710-733
Number of pages24
JournalJournal of Functional Analysis
Volume252
Issue number2
DOIs
Publication statusPublished - 15 Nov 2007
Externally publishedYes

Keywords

  • Boundary theory
  • Darning
  • Extension process
  • Feller measures
  • Jumping measure
  • Killing measure
  • Resolvent
  • Standard process
  • Time change
  • Weak duality

Fingerprint

Dive into the research topics of 'On Feller's boundary problem for Markov processes in weak duality'. Together they form a unique fingerprint.

Cite this

Chen, Z. Q., & Fukushima, M. (2007). On Feller's boundary problem for Markov processes in weak duality. Journal of Functional Analysis, 252(2), 710-733. https://doi.org/10.1016/j.jfa.2007.06.005