A necessary and sufficient condition for probabilistic continuity on a boundaryless compact Riemannian manifold

Abstract

We give a necessary and sufficient condition for the uniform convergence of random series of eigenfunctions on a boundaryless compact Riemannian manifold. Due to the lack of homogeneity of a compact manifold (by comparison with the case of compact groups studied by Marcus and Pisier), our proof relies on a suitable generalization of the Dudley-Fernique obtained via the theory of majorizing measures. As a consequence, we generalize an estimate of Burq and Lebeau about the supremum of a random eigenfunction. Finally, we prove that our results are universal w.r.t. the random variables (thus generalizing a result of Marcus and Pisier), w.r.t. compact submanifolds and w.r.t. the Riemannian structure of the underlying manifold.