The Semigroup Property, Cauchy Problem, and Gauss-Weierstrass Representation

On the strip R̲ⁿ × ]0, a[, we introduce a family of integral means M_b indexed by ]0, a]. Using these, we are able to give weak conditions for a temperature to have the semigroup property, for the Cauchy problem to have at most one solution corresponding to a given function f on R̲ⁿ × {0}, and for a temperature to be the Gauss-Weierstrass integral of some measure. Since all this is done from a unified point of view, the relationships between these properties are highlighted. The means M_b are very effective and easy to handle, because they exploit the semigroup property to the full.