Uniform convergence of stochastic semigroups

For stochastic C0-semigroups on L1-spaces there is a wealth of results that show strong convergence to an equilibrium as t → ∞, 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.