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The first concept is the one of shy coupling put forward in \\cite{Burdzy-Benjamini} and the second concept is the lower bound on the Ricci curvature and the connection with couplings made in \\cite{ReSt}.\n  The first construction is the shy coupling, the second one is a fixed-distance coupling and the third is a coupling in which the distance between the processes is a deterministic exponential function of time.\n  The result proved "},"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":"1210.7217","kind":"arxiv","version":7},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-10-26T19:11:49Z","cross_cats_sorted":[],"title_canon_sha256":"1e871e59af0c1db35bf7be0e9678aae69536d011f823e1dec33b0cf1e6c49979","abstract_canon_sha256":"f02e5f894d419fecd857b01af2db61c6d947560b0a2ef7cd9973dec13ad97c76"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:32:01.528774Z","signature_b64":"rcFfJn2M68Z+6lYXnRCICYTF0HBhQGKdT2+90mm676Sv1CLrzAv3t9LT0OH+77m2rqaVQyY1Bc9Bp2PrPayyCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"592333a6737925e8b7f760eac43bbd9b3e2a497c10e1dd03b3749ea16c5ea6f7","last_reissued_at":"2026-05-18T01:32:01.528350Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:32:01.528350Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Shy and Fixed-Distance Couplings of Brownian Motions on Manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Ionel Popescu, Mihai N. 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