  We consider two $C_0$-semigroups on function spaces or, more generally,
Banach lattices and give necessary and sufficient conditions for the orbits of
the first semigroup to dominate the orbits of the second semigroup for large
times. As an important special case we consider an $L^2$-space and self-adjoint
operators $A$ and $B$ which generate $C_0$-semigroups; in this situation we
give criteria for the existence of a time $t_1 \ge 0$ such that $e^{tB} \ge
e^{tA}$ for all subsequent times $t\ge t_1$.
  As a consequence of our abstract theory, we obtain many surprising insights
into the behaviour of various second and fourth order differential operators.
