  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.
