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

Greg Markowsky () and José Luis Palacios ()
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Greg Markowsky: Monash University
José Luis Palacios: The University of New Mexico

Methodology and Computing in Applied Probability, 2019, vol. 21, issue 3, 841-851

Abstract: 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.

Keywords: Birth-death chain; Green’s function; Markov chain; Electric resistance; Brownian motion; Local time; 60J10 (search for similar items in EconPapers)
Date: 2019
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DOI: 10.1007/s11009-017-9581-4

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