  We study Liouville theorems for the non-stationary Stokes equations in
exterior domains in $ \mathbb{R}^{n}$ under decay conditions for spatial
variables. As applications, we prove that the Stokes semigroup is a bounded
analytic semigroup on $L^{ \infty}_{ \sigma}$ of angle $ \pi/2$ for $n \geq 3$.
We also prove large time estimates for $n=2$ with zero net force.
