Positive harmonic functions vanishing on the boundary for the Laplacian in unbounded horn-shaped domains

Abstract

Denote points x ¯ ∈ R d + 1 \bar x \in {R^{d + 1}} , d ≥ 2 d \geq 2 , by x ¯ = ( ρ , θ , z ) \bar x = (\rho ,\theta ,z) , where ρ > 0 \rho > 0 , θ ∈ S d − 1 \theta \in {S^{d - 1}} , and z ∈ R z \in R . Let a : [ 0 , ∞ ) → ( 0 , ∞ ) a:[0,\infty ) \to (0,\infty ) be a nondecreasing C 2 {C^2} -function and define the "horn-shaped" domain Ω = { x ¯ = ( ρ , θ , z ) : | z | > a ( ρ ) } \Omega = \{ \bar x = (\rho ,\theta ,z):|z| > a(\rho )\} and its unit "cylinder" D = { x ¯ = ( ρ , θ , z ) ∈ Ω : ρ > 1 } D = \{ \bar x = (\rho ,\theta ,z) \in \Omega :\rho > 1\} . Under appropriate regularity conditions on a, we prove the following theorem: (i) If ∫ ∞ a ( ρ ) / ρ 2 d ρ = ∞ {\smallint ^\infty }a(\rho )/{\rho ^2}d\rho = \infty , then the Martin boundary at infinity for 1 2 Δ \frac {1}{2}\Delta in Ω \Omega is a single point, (ii) If ∫ ∞ a ( ρ ) / ρ 2 d ρ > ∞ {\smallint ^\infty }a(\rho )/{\rho ^2}d\rho > \infty , then the Martin boundary at infinity for 1 2 Δ \frac {1}{2}\Delta in Ω \Omega is homeomorphic to S d − 1 {S^{d - 1}} . More specifically, a sequence { ( ρ n , θ n , z n ) } n = 1 ∞ ⊂ Ω \{ ({\rho _n},{\theta _n},{z_n})\} _{n = 1}^\infty \subset \Omega satisfying lim n → ∞ ρ n = ∞ {\lim _{n \to \infty }}{\rho _n} = \infty is a Martin sequence if and only if lim n → ∞ θ n {\lim _{n \to \infty }}{\theta _n} exists on S d − 1 {S^{d - 1}} . From (i), it follows that the cone of positive harmonic functions in Ω \Omega vanishing continuously on ∂ Ω \partial \Omega is one-dimensional. From (ii), it follows easily that the cone of positive harmonic functions on Ω \Omega vanishing continuously on ∂ Ω \partial \Omega is generated by a collection of minimal elements which is homeomorphic to S d − 1 {S^{d - 1}} . In particular, the above result solves a problem stated by Kesten, who asked what the Martin boundary is for 1 2 Δ \frac {1}{2}\Delta in Ω \Omega in the case a ( ρ ) = 1 + ρ γ a(\rho ) = 1 + {\rho ^\gamma } , 0 > γ > 1 0 > \gamma > 1 . Our method of proof involves an analysis as ρ → ∞ \rho \to \infty of the exit distribution on ∂ D \partial D for Brownian motion starting from ( ρ , θ , z ) ∈ Ω (\rho ,\theta ,z) \in \Omega and conditioned to hit D before exiting Ω \Omega .