The inverse balayage problem for Markov chains

AbstractLet X be a Markov chain, let A be a finite sunset of its countable state space. let əA consist of states in A′ that can be reached in one step from A and let v be a prescribed probability measure on ∂A. In this paper we study the following inverse exit problem: describe and analyze the set M(v) of probability measures μ on A such that Pμ {X(T)ϵ·}=v(·) where T= inl{k: X(k)ϵ A′} is the first exit time from A. Characterizations are provided for elements of M(v), extreme points of M(v) and those measures in M(v) that are maximal with respect to a partial ordering induced by excessive functions. Potential theoretic aspects of the problem and one-dimensional birth and death processes are treated in detail, and examples are given that illustrate implications and limitations of the theory