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Let A be the W*-algebra, L1(E(0), u), where E(0) is a fnite set and u is a probability measure with full support. Let P: A->A be a completely positive unital map. In the present context, P is given by a stochastic matrix. We study the properties of P that are refected in the dilation theory developed by Muhly and Solel in Int. J. Math. 13, 2002. Let H be the Hilbert space L2(E(0), u) and let pi: A -> B(H) the representation of A given by multiplication. Form the Stinespring space H1. Then X is a W*-correspondence over P is expressed through a completely contractive representation T of X on H....