I will explain joint work with Amol Aggarwal and Promit Ghosal in which we construct the ASEP speed process that captures the joint behavior of perturbations of the asymmetric simple exclusion process under the basic coupling. In particular, consider an interacting particle system on $\mathbb{Z}$ in which each site $i\in\mathbb{Z}$ initially starts with a class $i$ particle. Independently and in continuous time, across each bond the neighboring sites swap their particles (and hence classes) with rate $p$ or $q$ (satisfying $p-q=1$) depending on the ordering of the classes. Letting $X_i(t)$ denote the location of the class $i$ particle at time $t$, we prove that $X_i(t)/t$ converges almost surely to a limit $U_i$ whose distribution is uniform on $[-1,1]$. The joint process $\{U_i\}_i$ defines the ASEP speed process.