The inverse balayage problem for Markov chains, part II

This paper continues the study of the inverse balayage problem for Markov chains. Let X be a Markov chain with state space A [up curve] B2, let v be a probability measure on B2 and let M(v) consist of probability measures [mu] on A whose X-balayage onto B2 is v. The faces of the compact, convex set M(v) are characterized. For fixed [mu] [epsilon] M(v) the set M([mu],v) of the measures [gimel, Hebrew] of the form [gimel, Hebrew](·) = P[mu]{X(S) [epsilon] ·}, where S is a randomized stopping time, is analyzed in detail. In particular, its extreme points and edge are explicitly identified. A naturally defined reversed chain X, for which v is an inverse balayage of [mu], is introduced and the relation between X and X is studied. The question of which [gimel, Hebrew] [epsilon] M([mu], v) admit a natural stopping time S[gimel, Hebrew] of X (not involving an independent randomization) such that [gimel, Hebrew](·) = P[mu]{X(S[gimel, Hebrew]) [epsilon] ·}, is shown to have rather different answers in discrete and continuous time. Illustrative examples are presented.