Discrete Heat Equation Morphisms

Abstract

We study heat kernels of locally finite graphs and discrete heat equation morphisms. These are combinatorial analogs to heat equation morphisms in Riemannian geometry (cf. E. Loubeau, [10]), parallel closely the discrete harmonic morphisms due to H. Urakawa, [13], and their properties are related to the initial value problem for the discrete heat equation. In applications we consider Hamming graphs (using the discrete Fourier calculus on Z2N), establish a heat kernel comparison theorem, and study the maps of e-nets induced by heat equation morphisms among two complete Riemannian manifolds.