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We use the fact that ${\\mathscr A}(x) = -{\\mathcal J}^{\\prime}(x)$ satisfies a weak monotonicity property to deduce uniform bounds in strong norms for solutions of the temporal, as well as of the spatio-temporal discretization of the problem. This weak monotonicity property then allows for the estimate $ \\underset{1 \\leq j \\leq J}\\sup {\\mathbb E}\\bigl[ \\Vert X_{t_j} - Y^j\\Vert_{{\\mathbb L}^2}^2\\bigr] \\leq C_{\\delta}(k^{1-\\delta} + h^2"},"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":"1705.09997","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-05-28T22:19:44Z","cross_cats_sorted":[],"title_canon_sha256":"2e5f00c94c10ca4da5ae771345c2f0977bb2286997d989f9917d10e18732f74e","abstract_canon_sha256":"f55a0c571549e132b31a6f9621305e87de24164f86e78b23ead979e8a49c684f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:38:17.898240Z","signature_b64":"pMTmjzy1/VbSjI84kAFeiflx6Makua7gc3f2WQ0fzlNZVsRs8phA2yvUa/9UIohceIYsPEPaIpePmgXo1qYYBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"12d7f86a6d906efcbbfd631139e5c225691810649eda2acf74b88ac20b866893","last_reissued_at":"2026-05-18T00:38:17.897394Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:38:17.897394Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Optimal Strong Rates of Convergence for a Space-Time Discretization of the Stochastic Allen-Cahn Equation with multiplicative noise","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Ananta K. 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This weak monotonicity property then allows for the estimate $ \\underset{1 \\leq j \\leq J}\\sup {\\mathbb E}\\bigl[ \\Vert X_{t_j} - Y^j\\Vert_{{\\mathbb L}^2}^2\\bigr] \\leq C_{\\delta}(k^{1-\\delta} + h^2"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.09997","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":"1705.09997","created_at":"2026-05-18T00:38:17.897539+00:00"},{"alias_kind":"arxiv_version","alias_value":"1705.09997v1","created_at":"2026-05-18T00:38:17.897539+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1705.09997","created_at":"2026-05-18T00:38:17.897539+00:00"},{"alias_kind":"pith_short_12","alias_value":"CLL7Q2TNSBXP","created_at":"2026-05-18T12:31:10.602751+00:00"},{"alias_kind":"pith_short_16","alias_value":"CLL7Q2TNSBXPZO75","created_at":"2026-05-18T12:31:10.602751+00:00"},{"alias_kind":"pith_short_8","alias_value":"CLL7Q2TN","created_at":"2026-05-18T12:31:10.602751+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/CLL7Q2TNSBXPZO75MMITTZOCEV","json":"https://pith.science/pith/CLL7Q2TNSBXPZO75MMITTZOCEV.json","graph_json":"https://pith.science/api/pith-number/CLL7Q2TNSBXPZO75MMITTZOCEV/graph.json","events_json":"https://pith.science/api/pith-number/CLL7Q2TNSBXPZO75MMITTZOCEV/events.json","paper":"https://pith.science/paper/CLL7Q2TN"},"agent_actions":{"view_html":"https://pith.science/pith/CLL7Q2TNSBXPZO75MMITTZOCEV","download_json":"https://pith.science/pith/CLL7Q2TNSBXPZO75MMITTZOCEV.json","view_paper":"https://pith.science/paper/CLL7Q2TN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1705.09997&json=true","fetch_graph":"https://pith.science/api/pith-number/CLL7Q2TNSBXPZO75MMITTZOCEV/graph.json","fetch_events":"https://pith.science/api/pith-number/CLL7Q2TNSBXPZO75MMITTZOCEV/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CLL7Q2TNSBXPZO75MMITTZOCEV/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CLL7Q2TNSBXPZO75MMITTZOCEV/action/storage_attestation","attest_author":"https://pith.science/pith/CLL7Q2TNSBXPZO75MMITTZOCEV/action/author_attestation","sign_citation":"https://pith.science/pith/CLL7Q2TNSBXPZO75MMITTZOCEV/action/citation_signature","submit_replication":"https://pith.science/pith/CLL7Q2TNSBXPZO75MMITTZOCEV/action/replication_record"}},"created_at":"2026-05-18T00:38:17.897539+00:00","updated_at":"2026-05-18T00:38:17.897539+00:00"}