Abstract: We discuss some topological properties of the persistent homology of subgaussian fields on compact Riemannian manifolds. For this class of processes, we are able to infer many desirable properties regarding the distribution of the persistence diagrams of the process, as well as some of its representations. We adopt the point of view of persistence measures and give out some results regarding the convergence of the empirical mean diagram towards the mean of the distribution (as defined by duality by the action of the persistence measures on measurable sets of $mathbb{R}^2$) and discuss the future perspectives of this subject.