{"paper":{"title":"An Elementary Proof for the Structure of Wasserstein Derivatives","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Cong Wu, Jianfeng Zhang","submitted_at":"2017-05-23T00:00:32Z","abstract_excerpt":"Let $F: \\mathbb{L}^2(\\Omega, \\mathbb{R}) \\to \\mathbb{R}$ be a law invariant and continuously Fr\\'echet differentiable mapping. Based on Lions \\cite{Lions}, Cardaliaguet \\cite{Cardaliaguet} (Theorem 6.2 and 6.5) proved that: \\bea \\label{Derivative} D F (\\xi) = g(\\xi), \\eea where $g: \\mathbb{R} \\to \\mathbb{R}$ is a deterministic function which depends only on the law of $\\xi$. See also Carmona \\& Delarue \\cite{CD} Section 5.2. In this short note we provide an elementary proof for this well known result. This note is part of our accompanying paper \\cite{WZ}, which deals with a more general situat"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.08046","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}