Potentials associated with recurrent Markov renewal processes

AbstractIn [1] and more recently in [2], Chapters III and VII, Spitzer constructs potentials for a particular class of recurrent Markov chains (M.C.'s), namely, the class of recurrent random walks on the n-dimensional lattice of integers (n = 1 or 2). In [3], [4], and [5], Kemeny and Snell construct a potential theory for arbitrary recurrent M.C.'s. Orey [6] has characterized potential kernels for recurrent M.C.'s. The potentials and kernels studied in these papers have properties which are analogous to those defined for transient M.C.'s. This paper extends the results of Kemeny and Snell to Markov renewal processes (MRP's).Section 1 contains a brief discussion of MRP's in general, a definition of the class of MRP's to be studied in this paper, and a summary of some needed relationships proved in other papers [7–10]. Section 2 contains some preliminary lemmas. In Section 3, a potential of functions is defined for the semi-Markov process (S-MP) associated with the MRP. The concept of a left-normal MRP is introduced and its relationship with a certain potential kernel is discussed. Section 4 contains applications of these results to the important case of a continuous parameter Markov chain (c.p.M.C.). Results are also obtained by means of “duality” for the potential of σ-finite signed measures defined for a c.p.M.C. In Section 5, a potential of functions is defined for an associated three-dimensional Markov process. Section 6 contains several miscellaneous results which have applications to previous sections.The MRP's considered are assumed to be irreducible, recurrent, nonlattice, and to satisfy hypothesis A (see below) unless otherwise specified. The nonlattice assumption causes no loss in generality, the lattice case (into which Kemeny and Snell's work falls) being handled by restricting t to take on only such values as are multiples of the span