  Archimedean copulas are popular in the world of multivariate modelling as a
result of their breadth, tractability, and flexibility. A. J. McNeil and J.
Ne\v{s}lehov\'a (2009) showed that the class of Archimedean copulas coincides
with the class of multivariate $\ell_1$-norm symmetric distributions. Building
upon their results, we introduce a class of multivariate Markov processes that
we call `Archimedean survival processes' (ASPs). An ASP is defined over a
finite time interval, is equivalent in law to a multivariate gamma process, and
its terminal value has an Archimedean survival copula. There exists a bijection
from the class of ASPs to the class of Archimedean copulas. We provide various
characterisations of ASPs, and a generalisation.
