Minimal Solutions of the Heat Equation and Uniqueness of the Positive Cauchy Problem on Homogeneous Spaces

The minimal positive solutions of the heat equation on A' X (-00, 7") are determined for X a homogeneous Riemannian space. A simple proof of uniqueness for the positive Cauchy problem on a homogeneous space is given using Choquet's theorem and the explicit form of these solutions. Introduction. A minimal solution of a linear elliptic or parabolic equation is a nonnegative solution u such that, whenever 0 < v < u is another solution, v = Xu with some constant 0 < À < 1. By Choquet's theorem all positive solutions are convex linear combinations of minimal solutions. It is well known that the minimal solutions of the heat equation Ah = u, of R" are the functions exp(||y\\2t + (x, y)) with y e R". In the present paper we generalise this result to a class of Riemannian manifolds with bounded geometry that includes all homogeneous Riemannian manifolds. Writing A for the Laplace-Beltrami operator, we show that all minimal solutions of the equation Aw = u, are of the form u(p, t) = e°"f(p) with/a minimal solution of the equation A/= af. We note that, in the particularly interesting case of noncompact Riemannian symmetric spaces, these functions/are explicitly known ([8], also [7]). The proof of our result consists of a simple application of Moser's parabolic Harnack inequality; it is inspired by the simple proof of the theorem of Karpelevic given by Y. Guivarc'h [7]. As we shall point out, the method applies also to a large class of parabolic equations on R". As an application of our main result we give a simple proof of the uniqueness of the positive Cauchy problem for the heat equation on homogeneous spaces. This proof avoids the growth estimates used in the classical arguments for parabolic equations on R". Received by the editors May 29, 1984 and, in revised form, August 14, 1984. 1980 Mathematics Subject Classification. Primary 35K15, 43A85; Secondary 31C12, 31C35.