I am searching for a reference about conditioning a Markov process in the sense of Doob, i.e. using h-transforms. My particular concern is to condition a discrete-time Markov Process on a possibly null-measure event, and I would need results such as conditions for uniqueness of extremal harmonic functions , maybe results related to Green's functions...

I found :

  • The (big) book of Doob, Classical potential theory and its probabilistic counterpart which is very general and would need a substantial investment to be understood.
  • The book of Rogers and Williams Diffusions, Markov processes and martingales, which is not very detailed, and mainly deals with the continuous time case.
  • Draft notes of A Bloemendal on the net http://www.math.harvard.edu/~alexb/rm/Doob.pdf
  • Some explanations in an article of O'Connel Conditioned random walks and RSK correspondence http://homepages.warwick.ac.uk/~masgas/pubs/noc03a.pdf

Anyone would have an other idea ?

  • $\begingroup$ One alternative reference : Denumerable markov chains, by Wolfgang Woess, 2009, if one is interested in homogeneous Markov chains. $\endgroup$ – saposcat Feb 8 '14 at 17:12
  • $\begingroup$ And better, the first volume of the book of Rogers and Williams, "Diffusions, Markov processes and martingales" actually summarizes Doob(and Martin, Hunt)'s theory. A smaller part of the second volume treats the case of h-transforms in the continuous case. $\endgroup$ – saposcat Feb 8 '14 at 17:15
  • $\begingroup$ In this brand new paper arxiv.org/pdf/1405.5157v1.pdf pag 19 they revise the historical definition of Doob's transform. Did you find other references? I'm also looking for them! $\endgroup$ – edwineveningfall Jun 6 '14 at 14:51

It's not a great answer (I've not enough reputation to comment) but it explains the concept without much mathematical garnish:


A book (page 242) Markov chains and mixing times - David A. Levin, Yuval Peres, Elizabeth L. Wilmer


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