Symmetry in the Green’s Function for Birth-death Chains

Greg Markowsky, José Luis Palacios

Research output: Contribution to journalArticleResearchpeer-review

Abstract

A symmetric relation in the probabilistic Green’s function for birth-death chains is explored. Two proofs are given, each of which makes use of the known symmetry of the Green’s functions in other contexts. The first uses as primary tool the local time of Brownian motion, while the second uses the reciprocity principle from electric network theory. We also show that the the second proof extends easily to cover birth-death chains (a.k.a. state-dependent random walks) on trees, and can be adapted in order to derive hitting times on trees.

Original languageEnglish
Pages (from-to)841-851
Number of pages11
JournalMethodology and Computing in Applied Probability
Volume21
Issue number3
DOIs
Publication statusPublished - 2019

Keywords

  • Birth-death chain
  • Brownian motion
  • Electric resistance
  • Green’s function
  • Local time
  • Markov chain

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