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We provide a Hunt type formula for the infinitesimal generators of Lévy process on the quantum groups $SU_q(N)$ and $U_q(N)$. In particular, we obtain a decomposition of such generators into a gaussian part and a `jump type' part determined by a linear functional that resembles the functional induced by the Lévy measure. The jump part on $SU_q(N)$ decomposes further into parts that live on the quantum subgroups $SU_q(n)$, $n\le N$. Like in the classical Hunt formula for locally compact Lie groups, the ingredients become unique once a certain projection is chosen. There are analogous result for $U_q(N)$.