Stopping times for recurrent Markov processes
John R. Baxter, R. V. Chacon · Illinois Journal of Mathematics · 1976
IntroductionLet {.} be a discrete-time Markov process with stationary transition prob- abilities, and let # be the distributions of o-Let z be a randomized stopping time, and let v be the distribution of .T hen say that/ can be balayaged to v, and write /v, or / v(z) to indicate the stopping time that effects the balayage.In this paper we consider the problem of giving an analytical ex- pression for E[z-I when # --* v(z).This problem has a well-known solution in the transient case.Let P be the transition operator of the process, and define the potential operator G--O=o pk.If #--* V(Z) then E[z] (l-v)G.This is the discrete-time analogue of the case of Brownian motion in dimension three or higher.In the Brownian motion case for dimension one or two the potential still exists as an operator on differences of probability measures, and the same formula remains valid.The discrete-time analogue of this situation would be a recurrent process such that the potential exists and such that tP" 0 as n for all probability measures t.For such a process the above formula is again valid, and we shall not deal with this case further.A general discussion of potential theory for recurrent processes is given in [9-1 and [11].In the present paper we wish to consider processes which are strongly re- current.It is assumed that tP" 2 as n c for all probability measures/, where 2 is an invariant probability measure, and that the potential operator G exists for differences of probability measures.It is then shown that/ v if and only if the negative part of (/ v)G is of the form d2, that is, absolutely continuous with respect to 2. Furthermore ess sup I1 min {E[z] #v(z)}.Thus a supremum has replaced the integral which occurred in the transient case. Balayage sequencesLet (S, ) be a measurable space and let {,, n 0, 1,...} be a discrete-time Markov process with state space S, having stationary transition probabilities p(x, A), where p is a Markov kernel on S x N. If/, is a measure on N,/,P will as usual denote the measure (2.1) #P(A) l(dx)p(x, A).