Long time heat diffusion on homogeneous trees

Abstract

Let X be a homogeneous tree of degree q + 1, q ≥ 2, L the Laplace operator of X and ht(x) the fundamental solution of the heat equation (∂t+L)u = 0 on X. We show that the heat kernel ht(x) is asymptotically concentrated in an annulus moving to infinity with finite speed R1 = (q−1)/(q+1). Asymptotic concentration of heat in the Lp norm is also investigated. 0. Introduction Let X be a homogeneous tree of degree q+ 1, that is, a connected graph with no loops in which every vertex is adjacent to q + 1 other vertices; in particular if q is equal to one, then the tree may be identified with the integers Z. We will write x ∼ y if x and y are adjacent. The tree is endowed with a natural distance function d, where d(x, y) is the number of edges in the unique path from x to y, and by a natural Laplace operator L defined by Lf(x) = 1 q + 1 ∑ y∼x [f(x)− f(y)]. If we denote by L(X) the Lebesgue spaces on X with respect to counting measure, then L is easily seen to be bounded on L(X) for every 1 ≤ p ≤ +∞, and selfadjoint on L(X), and therefore the heat operator Ht, which is spectrally defined on L(X) by Htf = ∫ σ2(L) edPλf ∀t ∈ (0,+∞) ∀f ∈ L(X), where σ2(L) denotes the L spectrum of L, and Pλ its spectral resolution, is also given by the series