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

TL;DR

This paper derives precise two-sided heat estimates for symmetric stable processes in complex horn-shaped regions, revealing nuanced behaviors of these processes and challenging existing assumptions about ultracontractivity and Varopoulos estimates.

Contribution

It provides the first comprehensive two-sided Dirichlet heat estimates for symmetric stable processes on unbounded horn-shaped regions, including cases lacking ultracontractivity.

Findings

Heat estimates are highly sensitive to the region's geometry.

Varopoulos-type estimates do not hold even with ultracontractivity.

Results apply to non-uniformly $C^{1,1}$ regions at infinity.

Abstract

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.