Arbitrarily finely divisible stochastic matrices

We introduce and study the class of arbitrarily finely divisible stochastic matrices ( AFD + -matrices): stochastic matrices that have a stochastic c -th root for infinitely many natural numbers c . This notion generalises the class of embeddable stochastic matrices. In particular, if A is a transition matrix for a Markov process over some time period, then arbitrary finely divisibility of A inside the set of stochastic matrices is the necessary and sufficient condition for the existence of a transition matrix corresponding to this Markov process over infinitesimally short periods. Our research explores the connection between the spectral properties of an AFD + -matrix A and the spectral properties of a limit point L of its stochastic roots. This connection, which is first formalised in the broader context of complex and real square matrices, poses restrictions on A assuming L is given. For example, if an AFD + -matrix A has a corresponding irreducible limit point L , then A has to be a circulant matrix. We identify all matrices that can be a limit point of stochastic roots for some non-singular AFD + -matrix. Further, we demonstrate a construction of a class of AFD + -matrices with a given limit point L from embeddable matrices. To illustrate these theoretical findings, we examine specific cases, including 2 × 2 matrices, 3 × 3 circulant matrices, and offer a complete characterisation of AFD + -matrices of rank-two.