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Under suitable conditions on the initial and final probability measures, we use convex duality \\`a la Bolza and Monge-Kantorovich theory to lift classical Hopf-Lax formulae from state space to Wasserst"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1705.05951","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-05-16T23:08:20Z","cross_cats_sorted":[],"title_canon_sha256":"15ebf3a01113927665a92096d75c51aa38bc8c601ff2fba749d99e7a9043fede","abstract_canon_sha256":"a7700eb3f44f68520bc31b1f5a4787b432e1c14a3b8856f2248be6cd7e5b1fa9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:42:38.748854Z","signature_b64":"KotrihikymnYZpowG34jpefWZgC+juAEXHg6r/kt5uvcmY2zLtDZan0yloRb0ScJoB+H9nVThDLF4lkvXlHNCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"175f7cb5316053c095dd1ebb568834e63b41a404d99d8a9e259516e4510e320b","last_reissued_at":"2026-05-18T00:42:38.748137Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:42:38.748137Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Optimal Ballistic Transport and Hopf-Lax Formulae on Wasserstein Space","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Nassif Ghoussoub","submitted_at":"2017-05-16T23:08:20Z","abstract_excerpt":"We investigate the optimal mass transport problem associated to the following \"ballistic\" cost functional on phase space $M\\times M^*$, $$ b_T(v, x):=\\inf\\{\\langle v, \\gamma (0)\\rangle +\\int_0^TL(\\gamma (t), {\\dot \\gamma}(t))\\, dt, \\gamma \\in C^1([0, T), M), \\gamma(T)=x\\}, $$ where $M=\\mathbb{R}^d$, $T>0$, and $L:M\\times M \\to \\mathbb{R}$ is a Lagrangian that is jointly convex in both variables. 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