Chapter 4. An analogue of $\mathrm{W}$ for discrete Markov chains
Joseph Najnudel, Bernard Roynette, Marc Yor, Joseph Najnudel, Bernard Roynette, Marc Yor · MSJ memoirs · 2009
Chapter 4.An analogue of W for discrete Markov chains. Introduction.In this chapter, we construct for Markov chains some σ-finite measures which enjoy similar properties as the measure W studied in Chapter 1. Very informally, these σ-finite measures are obtained by "conditioning a recurrent Markov process to be transient".Our construction applies to discrete versions of one-and two-dimensional Brownian motion, i.e. simple random walk on Z and Z 2 , but it can also be applied to a much larger class of Markov chains.This chapter is divided into three sections; in Section 4.1, we give the construction of the σ-finite measures mentioned above ; in Section 4.2, we study the main properties of these measures, and in Section 4.3, we study some examples in more details.4.1 Construction of the σ-finite measures (Q x , x ∈ E) 4.1.1Notation and hypothesis.Let E be a countable set, (X n ) n≥0 the canonical process on E N , (F n ) n≥0 its natural filtration, and F ∞ the σ-field generated by (X n ) n≥0 .Let us denote by (P x ) x∈E the family of probability measures on (E N , (F n ) n≥0 , F ∞ ) associated to a Markov chain (E x below denotes the expectation with respect to P x ) ; more precisely, we suppose there exist probability transitions (p y,z ) y,z∈E such that :