Factorization of Markov Chains

Existence of following factorization is proved:Here A is a stochastic or semi-stochastic (substohastic) d×d matrix (d); I is the unit matrix; B and C are nonnegative, upper and lower triangular matrices. B is a semistochastic matrix; the diagonal entries of C are 1. An exact information on properties of matrices B and C are obtained in particular cases. Some results on existence of invariant distribution x for Markov chains in the cases of absence or presence of sources g of walking particles are obtained using the factorization (F). These problems described by homogeneous or nonhomogeneous equation (I–A)x=g.