  The sequence of moments of a vector-valued random variable can characterize
its law. We study the analogous problem for path-valued random variables, that
is stochastic processes, by using so-called robust signature moments. This
allows us to derive a metric of maximum mean discrepancy type for laws of
stochastic processes and study the topology it induces on the space of laws of
stochastic processes. This metric can be kernelized using the signature kernel
which allows to efficiently compute it. As an application, we provide a
non-parametric two-sample hypothesis test for laws of stochastic processes.
