Stochastic R matrix for U_q(A_n ⁽¹⁾)

We show that the quantum R matrix for symmetric tensor representations of U_q(A_n ⁽¹⁾) satisfies the sum rule required for its stochastic interpretation under a suitable gauge. Its matrix elements at a special point of the spectral parameter are found to factorize into the form that naturally extends Povolotsky's local transition rate in the q-Hahn process for n=1. Based on these results we formulate new discrete and continuous time integrable Markov processes on a one-dimensional chain in terms of n species of particles obeying asymmetric stochastic dynamics. Bethe ansatz eigenvalues of the Markov matrices are also given.