Large time behavior of Dirichlet heat kernels on unbounded domains above the graph of a bounded Lipschitz function

Let D ⊆ Rd, d ≥ 2 be the unbounded domain above the graph of a bounded Lipschitz function. We study the asymptotic behavior of the transition density pD(t,x,y) of killed Brownian motions in D and show that limt → ∞ t(d+2)/2 pD(t,x,y) = C1u(x)u(y), where u is a minimal harmonic function corresponding to the Martin point at infinity and C1 is a positive constant