Potentials for Denumerable Markov Chains

Probabilistic generalizations of classical potential theory have been worked out by J. L. Doob [ l ] and G. A. Hunt [2]. Specialized to discrete processes, this provides a potential theory for transient Markov chains, analogous to the theory of the Newtonian potential. We provide a unified potential theory for all denumerable Markov chains; as applied to recurrent chains, the theory generalizes classical results on logarithmic potentials. We denote by P the transition matrix of a Markov chain with states the integers. DEFINITION. A function of states ƒ is called a right potential charge if it is integrable with respect to a given non-negative superregular measure a ( a P g a ) , and if the! limit g = l i m n ( / + P + • • • +P )f exists. Then g is called a right potential with charge/ . If fx is a finite measure and p = limn j i i ( / + P + • • • +P ) then v is called a left potential with charge JJL. The mapping ƒ—>ju, with ju» =ƒ,<*» is an isomorphism preserving all the interesting properties. E.g., ƒ is a right charge for P if and only if /x is a left charge for the so-called reverse chain. Since the reverse chains include all Markov chains, we automatically obtain for each theorem about functions a dual theorem about measures. In the transient case potentials can be represented by means of a positive potential operator G = I+P+P+ • • • , as g — Gf or v=iiG. In the recurrent case we show that af=0 is a necessary condition, or dually, /x must have total measure 0. In this case we have dual positive operators