Let X be a Markov chain, let A be a finite sunset of its countable state space. let [small schwa]A consist of states in A' that can be reached in one step from A and let v be a prescribed probability measure on [not partial differential]A. In this paper we study the following inverse exit problem: describe and analyze the set M(v) of probability measures [mu] on A such that P[mu] {X(T)[epsilon]·}=v(·) where T= inl{k: X(k)[epsilon] 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.
