  Suppose that $R$ is an excellent local domain with maximal ideal $m_R$. The
theory of multiplicities and mixed multiplicities of $m_R$-primary ideals
extends to (possibly non Noetherian) filtrations of $R$ by $m_R$-primary
ideals, and many of the classical theorems for $m_R$-primary ideals continue to
hold for filtrations. The celebrated theorems involving inequalities continue
to hold for filtrations, but the good conclusions that hold in the case of
equality for $m_R$-primary ideals do not hold for filtrations.
  In this article, we consider multiplicities and mixed multiplicities of $R$
by $m_R$-primary divisorial filtrations. We show that some important theorems
on equalities of multiplicities and mixed multiplicities of $m_R$-primary
ideals, which are not true in general for filtrations, are true for divisorial
filtrations. We prove that a theorem of Rees showing that if there is an
inclusion of $m_R$-primary ideals $I\subset I'$ with the same multiplicity then
$I$ and $I'$ have the same integral closure also holds for divisorial
filtrations. This theorem does not hold for arbitrary filtrations.
  We show that the Teissier Rees Sharp Katz theorem on equality in the
Minkowski inequality holds for divisorial filtrations in an excellent domain of
dimension two.
  We also show that the mixed multiplicities of divisorial filtrations are
anti-positive intersection products on a suitable normal scheme $X$
birationally dominating $R$, when $R$ is an algebraic local domain.
