Dissipative probability vector fields and generation of evolution semigroups in Wasserstein spaces

Abstract

We introduce and investigate a notion of multivalued $\lambda$-dissipative probability vector field (MPVF) in the Wasserstein space $\mathcal{P}_2(\mathsf X)$ of Borel probability measures on a Hilbert space $\mathsf X$. Taking inspiration from the theory of dissipative operators in Hilbert spaces and of Wasserstein gradient flows of geodesically convex functionals, we study local and global well posedness of evolution equations driven by dissipative MPVFs. Our approach is based on a measure-theoretic version of the Explicit Euler scheme, for which we prove novel convergence results with optimal error estimates under an abstract CFL stability condition, which do not rely on compactness arguments and also hold when $\mathsf X$ has infinite dimension. We characterize the limit solutions by a suitable Evolution Variational Inequality (EVI), inspired by the B\'enilan notion of integral solutions to dissipative evolutions in Banach spaces. Existence, uniqueness and stability of EVI solutions are then obtained under quite general assumptions, leading to the generation of a semigroup of nonlinear contractions.