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Euler and Betti curves of stochastic processes defined on a $d$-dimensionalcompact Riemannian manifold which are almost surely in a Sobolev space$W^{n,s}(X,\mathbb{R})$ (with $d<n$) are stable under perturbations of thedistributions of said processes in a Wasserstein metric. Moreover, Wassersteinstability is shown to hold for all $p>\frac{d}{n}$ for persistence diagramsstemming from functions in $W^{n,s}(X,\mathbb{R})$.

Euler and Betti curves are stable under Wasserstein deformations of distributions of stochastic processes