  In this paper, we consider symmetric $\alpha$-stable processes on (unbounded)
horn-shaped regions which are non-uniformly $C^{1,1}$ near infinity. By using
probabilistic approaches extensively, we establish two-sided Dirichlet heat
estimates of such processes for all time. The estimates are very sensitive with
respect to the reference function corresponding to each horn-shaped region. Our
results also cover the case that the associated Dirichlet semigroup is not
intrinsically ultracontractive. A striking observation from our estimates is
that, even when the associated Dirichlet semigroup is intrinsically
ultracontractive, the so-called Varopoulos-type estimates do not hold for
symmetric stable processes on horn-shaped regions.
