Decompositions of stochastic processes based on irreductible group representations

Let G be a topological compact group acting on some space Y. We study a decomposition of Y-indexed stochastic processes, based on the orthogonality relations between the characters of the irreducible representations of G. In the particular case of a Gaussian process with a G-invariant law, such a decomposition gives a very general explanation of a classic identity in law - between quadratic functionals of a Brownian bridge - due to Watson (1961). Several relations with Karhunen-Lo\`{e}ve expansions are discussed, and some applications and extensions are given - in particular related to Gaussian processes indexed by a torus.