  The local structure of $q$-Ornstein-Uhlenbeck processes and $q$-Brownian
motions are investigated, for all $q\in(-1,1)$. These are the classical Markov
processes corresponding to the noncommutative $q$-Gaussian processes. These
processes have discontinuous sample paths, and the local small jumps are
characterized by tangent processes. It is shown that for all $q\in(-1,1)$, the
tangent processes at inner domain are scaled Cauchy processes possibly with
drifts, and the tangent processes at the boundary of the domain are related to
the free $1/2$-stable law via Biane's construction.
