Metrics on the space of probability measures with applications

Abstract: Consider the space of probability measures on a complete and separable (Polish) metric space \$(E, d)\$. Call this space of measures \$P\$. One can place metrics on \$P\$ which are of varrying strengths: The Prohorov distance, the Wasserstein-p distance, and the common total variation distance. We introduce these metrics with a special view toward convergence of stochastic processes. Along the way we meet the Skorohod representation theorem, the notion of “coupling” random variables, and consider a hydrodynamic limit result (which leads to existence and uniqueness for a class of free boundary problems).