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

Euler and Betti curves of stochastic processes defined on a $d$-dimensional compact Riemannian manifold which are almost surely in a Sobolev space $W^{n,s}(X,\mathbb{R})$ (with $d\frac{d}{n}$ for persistence diagrams stemming from functions in $W^{n,s}(X,\mathbb{R})$.