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

• 成果类型: Article
• 署名作者: Cavagnari, Giulia; Savare, Giuseppe; Sodini, Giacomo Enrico
• 署名单位: Polytechnic University of Milan; Bocconi University; Bocconi University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-022-01148-7
• 发表日期: 2023
• 页码: 1087-1182
• 关键词: equations

摘要

We introduce and investigate a notion of multivalued lambda-dissipative probability vector field (MPVF) in the Wasserstein space P-2(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. We characterize the limit solutions by a suitable Evolution Variational Inequality (EVI), inspired by the Benilan 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.