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

Abstract

We introduce and investigate a notion of multivalued λ-dissipative probability vector field (MPVF) in the Wasserstein space P2(X) of Borel probability measures on a Hilbert space X. Taking inspiration from the theories of dissipative operators in Hilbert spaces and of Wasserstein gradient flows for 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 stability condition, which do not rely on compactness arguments and also hold when X has infinite dimension.