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These results are obtained under the condition that $\\int_{\\mathbb{R}^d}(1+|\\xi|^2)^{\\alpha-1}\\hat{f}(\\text{d} \\xi)<\\infty$ for some $\\alpha\\in(0,1]$, where $\\hat{f}$ is the spectral measure of t"},"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":"1607.03998","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-07-14T05:31:26Z","cross_cats_sorted":[],"title_canon_sha256":"9ef12d074f27ed6218653ac2a9b2d9aff2a8c6fd136308734ae832b649545c41","abstract_canon_sha256":"6cad061ee2160582c32ba0038cdfdd43e6e286201f65a3714f088b50107d050b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:11:04.343820Z","signature_b64":"cC3emYN4xpKah1SVJoHPs20z9LxU0ffBvZtt8RSb/zjtlnQaV7xLZx5WuRy1ddpZbN8e7GSMHATNLOCH6PmXBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1e24a38d9dc9aa434341ea9f88c038a73c254abe4e4c5fec95541aa513bb77e5","last_reissued_at":"2026-05-18T01:11:04.343365Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:11:04.343365Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Comparison principle for stochastic heat equation on $\\mathbb{R}^d$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Jingyu Huang, Le Chen","submitted_at":"2016-07-14T05:31:26Z","abstract_excerpt":"We establish the strong comparison principle and strict positivity of solutions to the following nonlinear stochastic heat equation on $\\mathbb{R}^d$ \\[ \\left(\\frac{\\partial }{\\partial t} -\\frac{1}{2}\\Delta \\right) u(t,x) = \\rho(u(t,x)) \\:\\dot{M}(t,x), \\] for measure-valued initial data, where $\\dot{M}$ is a spatially homogeneous Gaussian noise that is white in time and $\\rho$ is Lipschitz continuous. These results are obtained under the condition that $\\int_{\\mathbb{R}^d}(1+|\\xi|^2)^{\\alpha-1}\\hat{f}(\\text{d} \\xi)<\\infty$ for some $\\alpha\\in(0,1]$, where $\\hat{f}$ is the spectral measure of t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.03998","kind":"arxiv","version":1},"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":"1607.03998","created_at":"2026-05-18T01:11:04.343437+00:00"},{"alias_kind":"arxiv_version","alias_value":"1607.03998v1","created_at":"2026-05-18T01:11:04.343437+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1607.03998","created_at":"2026-05-18T01:11:04.343437+00:00"},{"alias_kind":"pith_short_12","alias_value":"DYSKHDM5ZGVE","created_at":"2026-05-18T12:30:12.583610+00:00"},{"alias_kind":"pith_short_16","alias_value":"DYSKHDM5ZGVEGQ2B","created_at":"2026-05-18T12:30:12.583610+00:00"},{"alias_kind":"pith_short_8","alias_value":"DYSKHDM5","created_at":"2026-05-18T12:30:12.583610+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/DYSKHDM5ZGVEGQ2B5KPYRQBYU4","json":"https://pith.science/pith/DYSKHDM5ZGVEGQ2B5KPYRQBYU4.json","graph_json":"https://pith.science/api/pith-number/DYSKHDM5ZGVEGQ2B5KPYRQBYU4/graph.json","events_json":"https://pith.science/api/pith-number/DYSKHDM5ZGVEGQ2B5KPYRQBYU4/events.json","paper":"https://pith.science/paper/DYSKHDM5"},"agent_actions":{"view_html":"https://pith.science/pith/DYSKHDM5ZGVEGQ2B5KPYRQBYU4","download_json":"https://pith.science/pith/DYSKHDM5ZGVEGQ2B5KPYRQBYU4.json","view_paper":"https://pith.science/paper/DYSKHDM5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1607.03998&json=true","fetch_graph":"https://pith.science/api/pith-number/DYSKHDM5ZGVEGQ2B5KPYRQBYU4/graph.json","fetch_events":"https://pith.science/api/pith-number/DYSKHDM5ZGVEGQ2B5KPYRQBYU4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/DYSKHDM5ZGVEGQ2B5KPYRQBYU4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/DYSKHDM5ZGVEGQ2B5KPYRQBYU4/action/storage_attestation","attest_author":"https://pith.science/pith/DYSKHDM5ZGVEGQ2B5KPYRQBYU4/action/author_attestation","sign_citation":"https://pith.science/pith/DYSKHDM5ZGVEGQ2B5KPYRQBYU4/action/citation_signature","submit_replication":"https://pith.science/pith/DYSKHDM5ZGVEGQ2B5KPYRQBYU4/action/replication_record"}},"created_at":"2026-05-18T01:11:04.343437+00:00","updated_at":"2026-05-18T01:11:04.343437+00:00"}