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The coefficients are merely measurable in $(\\omega,t)$ and can be unbounded and fully degenerate, that is, coefficients $a^{ij}$, $\\sigma^{ik}$ merely satisfy\n  \\begin{align}\n  \\label{abs only} \\left(\\alpha^{ij}(\\omega,t)\\right)_{d\\times d}"},"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":"1905.07545","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2019-05-18T07:29:14Z","cross_cats_sorted":[],"title_canon_sha256":"cc1b31d603e49c301f3269173773e2e0041cafaa26f956fdb7014e094d5e31d8","abstract_canon_sha256":"00aa6cee36472a6fe9bb6be2cb8db5b5d655cd37697fb7e6649a5e0b98854e84"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:45:49.387419Z","signature_b64":"ipZRPITvYMTwQT6K9s5gKHiynApJeAAE8Zx3DY1hJ6A4wPN0cqI4/Y+Tc034e2vhNM9Lokumjv1BrYMy9ZZIAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4788ea8df086ddb25315f9dcd4a06ffd632c4b98d0c35cedd23456ec47899092","last_reissued_at":"2026-05-17T23:45:49.386578Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:45:49.386578Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A sharp $L_p$-regularity result for second-order stochastic partial differential equations with unbounded and fully degenerate leading coefficients","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Ildoo Kim, Kyeong-Hun Kim","submitted_at":"2019-05-18T07:29:14Z","abstract_excerpt":"We present existence, uniqueness, and sharp regularity results of solution\n  to the stochastic partial differential equation (SPDE)\n  \\begin{align}\n  \\label{abs eqn}\n  du=(a^{ij}(\\omega,t)u_{x^ix^j}+f)dt + (\\sigma^{ik}(\\omega,t)u_{x^i}+g^k)dw^k_t, \\quad u(0,x)=u_0,\n  \\end{align}\n  where $\\{w^k_t:k=1,2,\\cdots\\}$ is a sequence of independent Brownian motions. The coefficients are merely measurable in $(\\omega,t)$ and can be unbounded and fully degenerate, that is, coefficients $a^{ij}$, $\\sigma^{ik}$ merely satisfy\n  \\begin{align}\n  \\label{abs only} \\left(\\alpha^{ij}(\\omega,t)\\right)_{d\\times d}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1905.07545","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":"1905.07545","created_at":"2026-05-17T23:45:49.386705+00:00"},{"alias_kind":"arxiv_version","alias_value":"1905.07545v1","created_at":"2026-05-17T23:45:49.386705+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1905.07545","created_at":"2026-05-17T23:45:49.386705+00:00"},{"alias_kind":"pith_short_12","alias_value":"I6EOVDPQQ3O3","created_at":"2026-05-18T12:33:18.533446+00:00"},{"alias_kind":"pith_short_16","alias_value":"I6EOVDPQQ3O3EUYV","created_at":"2026-05-18T12:33:18.533446+00:00"},{"alias_kind":"pith_short_8","alias_value":"I6EOVDPQ","created_at":"2026-05-18T12:33:18.533446+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/I6EOVDPQQ3O3EUYV7HONJIDP7V","json":"https://pith.science/pith/I6EOVDPQQ3O3EUYV7HONJIDP7V.json","graph_json":"https://pith.science/api/pith-number/I6EOVDPQQ3O3EUYV7HONJIDP7V/graph.json","events_json":"https://pith.science/api/pith-number/I6EOVDPQQ3O3EUYV7HONJIDP7V/events.json","paper":"https://pith.science/paper/I6EOVDPQ"},"agent_actions":{"view_html":"https://pith.science/pith/I6EOVDPQQ3O3EUYV7HONJIDP7V","download_json":"https://pith.science/pith/I6EOVDPQQ3O3EUYV7HONJIDP7V.json","view_paper":"https://pith.science/paper/I6EOVDPQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1905.07545&json=true","fetch_graph":"https://pith.science/api/pith-number/I6EOVDPQQ3O3EUYV7HONJIDP7V/graph.json","fetch_events":"https://pith.science/api/pith-number/I6EOVDPQQ3O3EUYV7HONJIDP7V/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/I6EOVDPQQ3O3EUYV7HONJIDP7V/action/timestamp_anchor","attest_storage":"https://pith.science/pith/I6EOVDPQQ3O3EUYV7HONJIDP7V/action/storage_attestation","attest_author":"https://pith.science/pith/I6EOVDPQQ3O3EUYV7HONJIDP7V/action/author_attestation","sign_citation":"https://pith.science/pith/I6EOVDPQQ3O3EUYV7HONJIDP7V/action/citation_signature","submit_replication":"https://pith.science/pith/I6EOVDPQQ3O3EUYV7HONJIDP7V/action/replication_record"}},"created_at":"2026-05-17T23:45:49.386705+00:00","updated_at":"2026-05-17T23:45:49.386705+00:00"}