HITTING TIME AND INVERSE PROBLEMS FOR MARKOV CHAINS VICTOR DE LA PENA, HENRYK GZYL AND PATRICK MCDONALD Abstract. Let Wn be a simple Markov chain on the integers. Suppose that Xn is a simple Markov chain on the integers whose transition prob- abilities coincide with those of Wn off a finite set. We prove that there is an M> 0 such that the Markov chain Wn and the joint distributions of first hitting time and first hitting place of Xn started at the origin for the sets {-M,M} and {-(M + 1), (M + 1)} algorithmically determine the transition probabilities of Xn. 1. Introduction Many problems arising in the natural sciences involve situations in which first hitting times for an unknown diffusion process driving particles in an inaccessible region are given, and from this data one seeks to determine properties of the underlying region and/or the particle dynamics (see [G] for a general reference; see [BC] for applications in neuroscience). As an illustrative model problem, consider a long thin tube containing a liquid whose diffusivity is known outside a given interval, say [−1, 1]. Suppose that particles are injected at the point 0 and first hitting time probabilities are given at a number of locations outside the inaccessible interval [−1, 1]. We ask: What properties of the diffusivity can be determined from the given data? In this note we study a discrete analog of the model diffusion problem sketched above. We formalize the problem as follows: Suppose that W n is a simple Markov chain on the integers, Z (ie the transition probabilities between two integers are nonzero if and only if the two integers are nearest neighbors). Suppose that D ⊂ Z is a finite subset of Z. Suppose that X n is a simple Markov chain whose transition probabilities coincide with those of W n outside of the subset D (we refer to such a Markov chain as a simple D-perturbation of W n ). The main result of this paper is the following Theorem 1.1. Let W n be a simple Markov chain on the integers. Suppose that D ⊂ Z is a finite set and that X n is a simple D-perturbation of W n . Let M = max{|i| : i ∈ D} +1. Then the Markov chain W n and the joint Received by the editors June 27, 2008. 2000 Mathematics Subject Classification. 60G40, 35R30, 39A12. Key words and phrases. inverse problems, Markov chains, first hitting times. 1