{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2007:7D5F25VBAD6WY7HOYG46P3JT4Y","short_pith_number":"pith:7D5F25VB","schema_version":"1.0","canonical_sha256":"f8fa5d76a100fd6c7ceec1b9e7ed33e6305329d61a5fddd3bdd24477bca20f3b","source":{"kind":"arxiv","id":"0710.5450","version":1},"attestation_state":"computed","paper":{"title":"Weak order for the discretization of the stochastic heat equation","license":"","headline":"","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Arnaud Debussche (IRMAR), Jacques Printems (LAMA)","submitted_at":"2007-10-29T15:06:29Z","abstract_excerpt":"In this paper we study the approximation of the distribution of $X_t$ Hilbert--valued stochastic process solution of a linear parabolic stochastic partial differential equation written in an abstract form as $$ dX_t+AX_t dt = Q^{1/2} d W_t, \\quad X_0=x \\in H, \\quad t\\in[0,T], $$ driven by a Gaussian space time noise whose covariance operator $Q$ is given. We assume that $A^{-\\alpha}$ is a finite trace operator for some $\\alpha>0$ and that $Q$ is bounded from $H$ into $D(A^\\beta)$ for some $\\beta\\geq 0$. It is not required to be nuclear or to commute with $A$. The discretization is achieved tha"},"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":"0710.5450","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.NA","submitted_at":"2007-10-29T15:06:29Z","cross_cats_sorted":["cs.NA"],"title_canon_sha256":"1c1b470ed741f417d2f843bdfb42901f135e4034480829aa56e93515a24894b3","abstract_canon_sha256":"5d011b4bbb89e22722be7b73982bc7337ce996c0b53644637acd89f9efdcb8da"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-03T22:06:03.043371Z","signature_b64":"b0C9LKfVbFCmg4daDj1ip6VkXzwGKhmPM9OoHf8fQvB/G4UgHitrnaKkmdjfmMRmUsqzq10TOMLYaQOD8H5dBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f8fa5d76a100fd6c7ceec1b9e7ed33e6305329d61a5fddd3bdd24477bca20f3b","last_reissued_at":"2026-06-03T22:06:03.042958Z","signature_status":"signed_v1","first_computed_at":"2026-06-03T22:06:03.042958Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Weak order for the discretization of the stochastic heat equation","license":"","headline":"","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Arnaud Debussche (IRMAR), Jacques Printems (LAMA)","submitted_at":"2007-10-29T15:06:29Z","abstract_excerpt":"In this paper we study the approximation of the distribution of $X_t$ Hilbert--valued stochastic process solution of a linear parabolic stochastic partial differential equation written in an abstract form as $$ dX_t+AX_t dt = Q^{1/2} d W_t, \\quad X_0=x \\in H, \\quad t\\in[0,T], $$ driven by a Gaussian space time noise whose covariance operator $Q$ is given. We assume that $A^{-\\alpha}$ is a finite trace operator for some $\\alpha>0$ and that $Q$ is bounded from $H$ into $D(A^\\beta)$ for some $\\beta\\geq 0$. It is not required to be nuclear or to commute with $A$. The discretization is achieved tha"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0710.5450","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/0710.5450/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"0710.5450","created_at":"2026-06-03T22:06:03.043020+00:00"},{"alias_kind":"arxiv_version","alias_value":"0710.5450v1","created_at":"2026-06-03T22:06:03.043020+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0710.5450","created_at":"2026-06-03T22:06:03.043020+00:00"},{"alias_kind":"pith_short_12","alias_value":"7D5F25VBAD6W","created_at":"2026-06-03T22:06:03.043020+00:00"},{"alias_kind":"pith_short_16","alias_value":"7D5F25VBAD6WY7HO","created_at":"2026-06-03T22:06:03.043020+00:00"},{"alias_kind":"pith_short_8","alias_value":"7D5F25VB","created_at":"2026-06-03T22:06:03.043020+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/7D5F25VBAD6WY7HOYG46P3JT4Y","json":"https://pith.science/pith/7D5F25VBAD6WY7HOYG46P3JT4Y.json","graph_json":"https://pith.science/api/pith-number/7D5F25VBAD6WY7HOYG46P3JT4Y/graph.json","events_json":"https://pith.science/api/pith-number/7D5F25VBAD6WY7HOYG46P3JT4Y/events.json","paper":"https://pith.science/paper/7D5F25VB"},"agent_actions":{"view_html":"https://pith.science/pith/7D5F25VBAD6WY7HOYG46P3JT4Y","download_json":"https://pith.science/pith/7D5F25VBAD6WY7HOYG46P3JT4Y.json","view_paper":"https://pith.science/paper/7D5F25VB","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0710.5450&json=true","fetch_graph":"https://pith.science/api/pith-number/7D5F25VBAD6WY7HOYG46P3JT4Y/graph.json","fetch_events":"https://pith.science/api/pith-number/7D5F25VBAD6WY7HOYG46P3JT4Y/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7D5F25VBAD6WY7HOYG46P3JT4Y/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7D5F25VBAD6WY7HOYG46P3JT4Y/action/storage_attestation","attest_author":"https://pith.science/pith/7D5F25VBAD6WY7HOYG46P3JT4Y/action/author_attestation","sign_citation":"https://pith.science/pith/7D5F25VBAD6WY7HOYG46P3JT4Y/action/citation_signature","submit_replication":"https://pith.science/pith/7D5F25VBAD6WY7HOYG46P3JT4Y/action/replication_record"}},"created_at":"2026-06-03T22:06:03.043020+00:00","updated_at":"2026-06-03T22:06:03.043020+00:00"}