Symmetries of the asymptotic dynamics of random compositions of equivariant maps

Let be a compact Lie group and let denote the connected component of the identity of . Suppose f is a map equivariant with respect to . We consider perturbations of f which are modelled by random compositions of equivariant maps which are close to f pointwise. We show that under mild assumptions on the distribution governing the choice of maps any invariant measure for the resulting Markov process is invariant and absolutely continuous with respect to Lebesgue measure. Thus observations on the asymptotic dynamics of the perturbed system will have symmetry.