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We first give a representation of $X$ using excursions of a continuous state branching process and Arratia's coalescing Brownian flow. For any nonnegative, nondecreasing and right continuous function $g$, put\n  \\tau:=\\sup \\{t\\geq 0: X_t([-g(t),g(t)])>0 \\}. We prove that $\\bP\\{\\tau=\\infty\\}=0$ or 1 according as the integral $\\int_1^\\infty g(t)t^{-1-1/\\beta} dt$ is finite or infinite."},"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":"0911.0774","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2009-11-04T10:40:30Z","cross_cats_sorted":[],"title_canon_sha256":"e658349372d3e5e21425a3137e077893a76faac2a44da3311ec634a219dd6e93","abstract_canon_sha256":"a2adffd0b05a0a08c539ca6afcdac4936d69d89b8f34245f3cd7ae298167d75b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:05:16.064013Z","signature_b64":"EwywOymBwhKERldhfzpeSapu+01dFzTiiO6iFtenjk9AofZE0BNkRkUfncTtA5VchmEaD5m2kP05ImHQHG9+CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4abac60e4adec5db9e55f10a53a2ab547c794bc930b6f6bd3b472b4a9de3acb7","last_reissued_at":"2026-05-18T04:05:16.063548Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:05:16.063548Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"An integral test on time dependent local extinction for super-coalescing Brownian motion with Lebesgue initial measure","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Hui He, Xiaowen Zhou, Zenghu Li","submitted_at":"2009-11-04T10:40:30Z","abstract_excerpt":"This paper concerns the almost sure time dependent local extinction behavior for super-coalescing Brownian motion $X$ with $(1+\\beta)$-stable branching and Lebesgue initial measure on $\\bR$. We first give a representation of $X$ using excursions of a continuous state branching process and Arratia's coalescing Brownian flow. For any nonnegative, nondecreasing and right continuous function $g$, put\n  \\tau:=\\sup \\{t\\geq 0: X_t([-g(t),g(t)])>0 \\}. 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