{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:2OTHBU7CFPJQZIFYIBSJ56B3QB","short_pith_number":"pith:2OTHBU7C","canonical_record":{"source":{"id":"1906.08638","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-06-20T14:09:38Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"deca26f64c97ae9f17f11f0e64f4326db5fbcf1a47f407629aef5f7d51b263d1","abstract_canon_sha256":"663164b8f36cd420230a1d1bc43996113b75b8f910e05d76ef5adfae8785d88b"},"schema_version":"1.0"},"canonical_sha256":"d3a670d3e22bd30ca0b840649ef83b806cedaff1ab650d3250be91450928c1fe","source":{"kind":"arxiv","id":"1906.08638","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1906.08638","created_at":"2026-05-17T23:42:50Z"},{"alias_kind":"arxiv_version","alias_value":"1906.08638v1","created_at":"2026-05-17T23:42:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1906.08638","created_at":"2026-05-17T23:42:50Z"},{"alias_kind":"pith_short_12","alias_value":"2OTHBU7CFPJQ","created_at":"2026-05-18T12:33:07Z"},{"alias_kind":"pith_short_16","alias_value":"2OTHBU7CFPJQZIFY","created_at":"2026-05-18T12:33:07Z"},{"alias_kind":"pith_short_8","alias_value":"2OTHBU7C","created_at":"2026-05-18T12:33:07Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:2OTHBU7CFPJQZIFYIBSJ56B3QB","target":"record","payload":{"canonical_record":{"source":{"id":"1906.08638","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-06-20T14:09:38Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"deca26f64c97ae9f17f11f0e64f4326db5fbcf1a47f407629aef5f7d51b263d1","abstract_canon_sha256":"663164b8f36cd420230a1d1bc43996113b75b8f910e05d76ef5adfae8785d88b"},"schema_version":"1.0"},"canonical_sha256":"d3a670d3e22bd30ca0b840649ef83b806cedaff1ab650d3250be91450928c1fe","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:42:50.019723Z","signature_b64":"R8mgJVJb+9hL6T53/6IZZaArGv04RaqInfoeB45LE3Ut9YiTsrXjsSDxzKSSyC2uqTxhO2xA1/8rTmdPEmkqAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d3a670d3e22bd30ca0b840649ef83b806cedaff1ab650d3250be91450928c1fe","last_reissued_at":"2026-05-17T23:42:50.019099Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:42:50.019099Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1906.08638","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:42:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"N1RPxoDRRYRnH6P475cFm+Cdpa5b8DuQ9/s8TsDWvCkohGwcmTEJKhpBTQ/R0uqodDbgnJ3uwVu5YlICzb8XAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T04:41:10.441192Z"},"content_sha256":"8685395e41c8faf09c2c1bbd4066fb4a9d4570264fbece68c782bdd19f74c047","schema_version":"1.0","event_id":"sha256:8685395e41c8faf09c2c1bbd4066fb4a9d4570264fbece68c782bdd19f74c047"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:2OTHBU7CFPJQZIFYIBSJ56B3QB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The stochastic nonlinear Schr\\\"odinger equation in unbounded domains and manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.AP","authors_text":"Fabian Hornung","submitted_at":"2019-06-20T14:09:38Z","abstract_excerpt":"In this article, we construct a global martingale solution to a general nonlinear Schr\\\"{o}dinger equation with linear multiplicative noise in the Stratonovich form. Our framework includes many examples of spatial domains like $\\mathbb{R}^d$, non-compact Riemannian manifolds, and unbounded domains in $\\mathbb{R}^d$ with different boundary conditions. The initial value belongs to the energy space $H^1$ and we treat subcritical focusing and defocusing power nonlinearities. The proof is based on an approximation technique which makes use of spectral theoretic methods and an abstract Littlewood-Pa"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1906.08638","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:42:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hkkWHG48gM4Z+cYWnwmiQ4jFZdOt0WXTutSEUH6nBCkdCAzd++GaMbsNL0xYfQuh7FxLmG2ppylKqYXIlQeDCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T04:41:10.441674Z"},"content_sha256":"dd5e42779d358f977da3c73c5ba4b7e61839f21320acd34c7bbe62445109735b","schema_version":"1.0","event_id":"sha256:dd5e42779d358f977da3c73c5ba4b7e61839f21320acd34c7bbe62445109735b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/2OTHBU7CFPJQZIFYIBSJ56B3QB/bundle.json","state_url":"https://pith.science/pith/2OTHBU7CFPJQZIFYIBSJ56B3QB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/2OTHBU7CFPJQZIFYIBSJ56B3QB/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-26T04:41:10Z","links":{"resolver":"https://pith.science/pith/2OTHBU7CFPJQZIFYIBSJ56B3QB","bundle":"https://pith.science/pith/2OTHBU7CFPJQZIFYIBSJ56B3QB/bundle.json","state":"https://pith.science/pith/2OTHBU7CFPJQZIFYIBSJ56B3QB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/2OTHBU7CFPJQZIFYIBSJ56B3QB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:2OTHBU7CFPJQZIFYIBSJ56B3QB","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"663164b8f36cd420230a1d1bc43996113b75b8f910e05d76ef5adfae8785d88b","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-06-20T14:09:38Z","title_canon_sha256":"deca26f64c97ae9f17f11f0e64f4326db5fbcf1a47f407629aef5f7d51b263d1"},"schema_version":"1.0","source":{"id":"1906.08638","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1906.08638","created_at":"2026-05-17T23:42:50Z"},{"alias_kind":"arxiv_version","alias_value":"1906.08638v1","created_at":"2026-05-17T23:42:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1906.08638","created_at":"2026-05-17T23:42:50Z"},{"alias_kind":"pith_short_12","alias_value":"2OTHBU7CFPJQ","created_at":"2026-05-18T12:33:07Z"},{"alias_kind":"pith_short_16","alias_value":"2OTHBU7CFPJQZIFY","created_at":"2026-05-18T12:33:07Z"},{"alias_kind":"pith_short_8","alias_value":"2OTHBU7C","created_at":"2026-05-18T12:33:07Z"}],"graph_snapshots":[{"event_id":"sha256:dd5e42779d358f977da3c73c5ba4b7e61839f21320acd34c7bbe62445109735b","target":"graph","created_at":"2026-05-17T23:42:50Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"In this article, we construct a global martingale solution to a general nonlinear Schr\\\"{o}dinger equation with linear multiplicative noise in the Stratonovich form. Our framework includes many examples of spatial domains like $\\mathbb{R}^d$, non-compact Riemannian manifolds, and unbounded domains in $\\mathbb{R}^d$ with different boundary conditions. The initial value belongs to the energy space $H^1$ and we treat subcritical focusing and defocusing power nonlinearities. The proof is based on an approximation technique which makes use of spectral theoretic methods and an abstract Littlewood-Pa","authors_text":"Fabian Hornung","cross_cats":["math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-06-20T14:09:38Z","title":"The stochastic nonlinear Schr\\\"odinger equation in unbounded domains and manifolds"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1906.08638","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:8685395e41c8faf09c2c1bbd4066fb4a9d4570264fbece68c782bdd19f74c047","target":"record","created_at":"2026-05-17T23:42:50Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"663164b8f36cd420230a1d1bc43996113b75b8f910e05d76ef5adfae8785d88b","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-06-20T14:09:38Z","title_canon_sha256":"deca26f64c97ae9f17f11f0e64f4326db5fbcf1a47f407629aef5f7d51b263d1"},"schema_version":"1.0","source":{"id":"1906.08638","kind":"arxiv","version":1}},"canonical_sha256":"d3a670d3e22bd30ca0b840649ef83b806cedaff1ab650d3250be91450928c1fe","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d3a670d3e22bd30ca0b840649ef83b806cedaff1ab650d3250be91450928c1fe","first_computed_at":"2026-05-17T23:42:50.019099Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:42:50.019099Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"R8mgJVJb+9hL6T53/6IZZaArGv04RaqInfoeB45LE3Ut9YiTsrXjsSDxzKSSyC2uqTxhO2xA1/8rTmdPEmkqAw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:42:50.019723Z","signed_message":"canonical_sha256_bytes"},"source_id":"1906.08638","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8685395e41c8faf09c2c1bbd4066fb4a9d4570264fbece68c782bdd19f74c047","sha256:dd5e42779d358f977da3c73c5ba4b7e61839f21320acd34c7bbe62445109735b"],"state_sha256":"071e3cc0481dcb5a91569abc983494bdaebcf954d3cf1b6312da492f5fa5fe9b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2o6auXffGhdq+KUxyPyVXiYla2lB0v81ipbiYZRWS0vwgWMojBiHocnRvHG8q2Pp4nBB6P/wg21DI28uNt8bBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T04:41:10.444997Z","bundle_sha256":"b06a535528b46e07c685856f645d3bfdcb0800de833861ace7f8102baef3b1ab"}}