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It is well known that the Bakry-Emery curvature is bounded below by a positive constant $\\ll>0$ if and only if $$W_p(\\mu_1P_t, \\mu_2P_t)\\le \\e^{-\\ll t} W_p (\\mu_1,\\mu_2),\\ \\ t\\ge 0, p\\ge 1 $$ holds for all probability measures $\\mu_1$ and $\\mu_2$ on $M$, where $W_p$ is the $L^p$ Wasserstein distance induced by the Riemannian distance. In this paper, we prove the exponential contraction $$W_p(\\mu_1P_t, \\mu_2P_t)\\le c\\e"},"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":"1603.05749","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-03-18T01:58:21Z","cross_cats_sorted":[],"title_canon_sha256":"ed6163fb6a3de21f3b0ab0bf34c1f43a8d1ed61ea2d4a036824bab5c1ea599fc","abstract_canon_sha256":"d15ff72cc566c3ebba7a84f0f9bfd9b18fad6783e5004e7d66e21a2b817cb26b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:57:04.670361Z","signature_b64":"0zVWEopW3p8jIjAfG2T2wTZ2rknpttuinIryt0LmMiu5r1SapNNg3doqUlQUIvX9dF7rRBaIBVm/l1K3XKpLDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"04ef7da3914d13e0d315b122e948dc053fc2b99a2efcf3cce750129a66f4364d","last_reissued_at":"2026-05-18T00:57:04.669854Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:57:04.669854Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Exponential Contraction in Wasserstein Distances for Diffusion Semigroups with Negative Curvature","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Feng-Yu Wang","submitted_at":"2016-03-18T01:58:21Z","abstract_excerpt":"Let $P_t$ be the (Neumann) diffusion semigroup $P_t$ generated by a weighted Laplacian on a complete connected Riemannian manifold $M$ without boundary or with a convex boundary. 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