  We show that the quantum $R$ matrix for symmetric tensor representations of
$U_q(A^{(1)}_n)$ 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.
