  For stochastic $C_0$-semigroups on $L^1$-spaces there is wealth of results
that show strong convergence to an equilibrium as $t \to \infty$, given that
the semigroup contains a partial integral operator. This has plenty of
applications to transport equations and in mathematical biology. However, up to
now partial integral operators do not play a prominent role in theorems which
yield uniform convergence of the semigroup rather than only strong convergence.
  In this article we prove that, for irreducible stochastic semigroups, uniform
convergence to an equilibrium is actually equivalent to being partially
integral and uniformly mean ergodic. In addition to this Tauberian theorem, we
also show that our semigroup is uniformly convergent if and only if it is
partially integral and the dual semigroup satisfies a certain irreducibility
condition. Our proof is based on a uniform version of a lower bound theorem of
Lasota and Yorke, which we combine with various techniques from Banach lattice
theory.
