Stochastic $R$ matrix for $U_q(A^{(1)}_n)$

Abstract: 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.