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Here we study the last zero crossing of $ ^{\\mu_1}_{\\mu_2}\\!I(t) $ and for this purpose we derive the last zero-crossing distribution of the drifted Brownian motion. We derive also the joint distribution of the last zero crossing before $ t $ and of the first passage time through the zero level of a Brownian motion with drift $ \\mu $ after $ t $. All these results permit us to derive explicit"},"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":"1803.00877","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-03-02T15:07:52Z","cross_cats_sorted":[],"title_canon_sha256":"41d5cef716c404c802bbc5d0f98518935546262cfe75c55f08f4e57ca3cf9d06","abstract_canon_sha256":"c2e5e260aa4eabee707a1b9437396c5e8abc663e1b0d3bd70b84ae786da6bef7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:44:10.275327Z","signature_b64":"ggRp3MJoY8JeKPZOHNAaam0stlp7JxS6fHvljUs6mAGhCVOgMnUkwT9WhgZcya6U3aMe3zeqqlNGJZNYa08oCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a0558740b052aec5fe20d8cf8b007abe4e37c38d12ef64a048688c211e2c46ab","last_reissued_at":"2026-05-17T23:44:10.274632Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:44:10.274632Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The last zero crossing of an iterated Brownian motion with drift","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Enzo Orsingher, Francesco Iafrate","submitted_at":"2018-03-02T15:07:52Z","abstract_excerpt":"In this paper we consider the iterated Brownian motion $ ^{\\mu_1}_{\\mu_2}\\!I(t) = B_1^{\\mu_1} ( | B_{2}^{\\mu_2} (t)|) $ where $B_j^{\\mu_j} , j=1,2$ are two independent Brownian motions with drift $\\mu_j$. Here we study the last zero crossing of $ ^{\\mu_1}_{\\mu_2}\\!I(t) $ and for this purpose we derive the last zero-crossing distribution of the drifted Brownian motion. We derive also the joint distribution of the last zero crossing before $ t $ and of the first passage time through the zero level of a Brownian motion with drift $ \\mu $ after $ t $. 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