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We show that\n  (i) If $0<\\alpha<1$ then with positive probability, $u(t,x)=0$ for some $(t,x)$.\n  (ii) If $\\alpha>3$ then with probability one, $u(t,x)\\ne0$ for all $(t,x)$."},"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":"1802.09487","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-02-26T18:10:05Z","cross_cats_sorted":[],"title_canon_sha256":"a5cb46867aca34f021af673c53f9c114ea0b78ac28ef2fffe51cf92734fad634","abstract_canon_sha256":"f8226df3e316a7864d924f1b7c75598fd3b396365d73ee7575534014d28c2928"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:53:09.688959Z","signature_b64":"JCVHjXtp789ocqm0Tt4Va0hgpLJSqKTBFTgXEnoNSFU37Bb3Ll5zK0e5QcurBwqFGnwdkdAxV53OqGzDegrFBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c389f920bc5ca2be0ba10fab7e920bdccf83f1ced74fef532431c2de9c3a6a7c","last_reissued_at":"2026-05-17T23:53:09.688271Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:53:09.688271Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Can the Stochastic Wave Equation with Strong Drift Hit Zero?","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Carl Mueller, Kevin Lin","submitted_at":"2018-02-26T18:10:05Z","abstract_excerpt":"We study the stochastic wave equation with multiplicative noise and singular drift: \\[ \\partial_tu(t,x)=\\Delta u(t,x)+u^{-\\alpha}(t,x)+g(u(t,x))\\dot{W}(t,x) \\] where $x$ lies in the circle $\\mathbf{R}/J\\mathbf{Z}$ and $u(0,x)>0$. We show that\n  (i) If $0<\\alpha<1$ then with positive probability, $u(t,x)=0$ for some $(t,x)$.\n  (ii) If $\\alpha>3$ then with probability one, $u(t,x)\\ne0$ for all $(t,x)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.09487","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1802.09487","created_at":"2026-05-17T23:53:09.688398+00:00"},{"alias_kind":"arxiv_version","alias_value":"1802.09487v2","created_at":"2026-05-17T23:53:09.688398+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1802.09487","created_at":"2026-05-17T23:53:09.688398+00:00"},{"alias_kind":"pith_short_12","alias_value":"YOE7SIF4LSRL","created_at":"2026-05-18T12:33:04.347982+00:00"},{"alias_kind":"pith_short_16","alias_value":"YOE7SIF4LSRL4C5B","created_at":"2026-05-18T12:33:04.347982+00:00"},{"alias_kind":"pith_short_8","alias_value":"YOE7SIF4","created_at":"2026-05-18T12:33:04.347982+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/YOE7SIF4LSRL4C5BB6VX5EQL3T","json":"https://pith.science/pith/YOE7SIF4LSRL4C5BB6VX5EQL3T.json","graph_json":"https://pith.science/api/pith-number/YOE7SIF4LSRL4C5BB6VX5EQL3T/graph.json","events_json":"https://pith.science/api/pith-number/YOE7SIF4LSRL4C5BB6VX5EQL3T/events.json","paper":"https://pith.science/paper/YOE7SIF4"},"agent_actions":{"view_html":"https://pith.science/pith/YOE7SIF4LSRL4C5BB6VX5EQL3T","download_json":"https://pith.science/pith/YOE7SIF4LSRL4C5BB6VX5EQL3T.json","view_paper":"https://pith.science/paper/YOE7SIF4","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1802.09487&json=true","fetch_graph":"https://pith.science/api/pith-number/YOE7SIF4LSRL4C5BB6VX5EQL3T/graph.json","fetch_events":"https://pith.science/api/pith-number/YOE7SIF4LSRL4C5BB6VX5EQL3T/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/YOE7SIF4LSRL4C5BB6VX5EQL3T/action/timestamp_anchor","attest_storage":"https://pith.science/pith/YOE7SIF4LSRL4C5BB6VX5EQL3T/action/storage_attestation","attest_author":"https://pith.science/pith/YOE7SIF4LSRL4C5BB6VX5EQL3T/action/author_attestation","sign_citation":"https://pith.science/pith/YOE7SIF4LSRL4C5BB6VX5EQL3T/action/citation_signature","submit_replication":"https://pith.science/pith/YOE7SIF4LSRL4C5BB6VX5EQL3T/action/replication_record"}},"created_at":"2026-05-17T23:53:09.688398+00:00","updated_at":"2026-05-17T23:53:09.688398+00:00"}