We study both the classical and the quantum dynamical system
associated to the (non-)linear Klein-Gordon equation on the hyperboloid.
The Fourier-Helgason transform provides a detailed and very explicit
description of the quantum fields describing free massive scalar bosons on
the de Sitter space. The interacting fields result from Markov processes on
the Euclidean sphere, defined on a probability space with a non-Gaussian
measure, and an Osterwalder-Schrader type reconstruction theorem. We also
prove that there exists a unique de Sitter vacuum state for all positive
coupling constants.