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

Two-sided Dirichlet heat estimates of symmetric stable processes on horn-shaped regions