As an extension of the work of Denzel, Kemeny, and Snell on the excessive functions of a continuous time Markov chain, this paper introduces the concept of reconstructability in two forms. First, there is reconstructability from the class of excessive functions, where it is seen that the transition matrix for a transient chain with a finite atomic exit boundary can be written down knowing only the membership of its class of excessive functions. A similar result is true, with the transient condition dropped, for reconstructability from the characteristic operator, based on a natural extension to the boundary of the operator corresponding to the initial derivative matrix.
"The Reconstructability of Markov Chains." Ann. Probab. 4 (1) 127 - 132, February, 1976. https://doi.org/10.1214/aop/1176996191