{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:P6S4L3NSOYDTE35L3LM6RHNRTQ","short_pith_number":"pith:P6S4L3NS","canonical_record":{"source":{"id":"2605.20553","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.AP","submitted_at":"2026-05-19T23:10:23Z","cross_cats_sorted":[],"title_canon_sha256":"965d3e65a5d042116c15da2ad6887cf849419ff32ba97c0df6b19db8a66a1037","abstract_canon_sha256":"d6288de461aa746d4a70e09b6d0665237cc8ba6e58375f11c07e74174f1e6238"},"schema_version":"1.0"},"canonical_sha256":"7fa5c5edb27607326fabdad9e89db19c3ef82d65c1e153bc71d0703a45f64b54","source":{"kind":"arxiv","id":"2605.20553","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.20553","created_at":"2026-05-21T01:04:42Z"},{"alias_kind":"arxiv_version","alias_value":"2605.20553v1","created_at":"2026-05-21T01:04:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.20553","created_at":"2026-05-21T01:04:42Z"},{"alias_kind":"pith_short_12","alias_value":"P6S4L3NSOYDT","created_at":"2026-05-21T01:04:42Z"},{"alias_kind":"pith_short_16","alias_value":"P6S4L3NSOYDTE35L","created_at":"2026-05-21T01:04:42Z"},{"alias_kind":"pith_short_8","alias_value":"P6S4L3NS","created_at":"2026-05-21T01:04:42Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:P6S4L3NSOYDTE35L3LM6RHNRTQ","target":"record","payload":{"canonical_record":{"source":{"id":"2605.20553","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.AP","submitted_at":"2026-05-19T23:10:23Z","cross_cats_sorted":[],"title_canon_sha256":"965d3e65a5d042116c15da2ad6887cf849419ff32ba97c0df6b19db8a66a1037","abstract_canon_sha256":"d6288de461aa746d4a70e09b6d0665237cc8ba6e58375f11c07e74174f1e6238"},"schema_version":"1.0"},"canonical_sha256":"7fa5c5edb27607326fabdad9e89db19c3ef82d65c1e153bc71d0703a45f64b54","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-21T01:04:42.270446Z","signature_b64":"ZF049oiytufpFHLzjDN1/+GOl1D11DMyc1IrfvGRlEYNijOiw2NKqFPxEvsQA0VOAv76StpIFlMjfvQamq9DBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7fa5c5edb27607326fabdad9e89db19c3ef82d65c1e153bc71d0703a45f64b54","last_reissued_at":"2026-05-21T01:04:42.269746Z","signature_status":"signed_v1","first_computed_at":"2026-05-21T01:04:42.269746Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2605.20553","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":"2026-05-21T01:04:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mJikKEU4PhYBG23sj6F7My203wel5z3XfMZJgx7kRN/Uhfuoj+oPjiMmtjsYTtKqSu2m8ExBSGFtLWLw+PMuBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-24T20:43:58.925911Z"},"content_sha256":"89cb5e31b6873c00285d26b950f77fc182b45ff21bce0bf028d9fd8f55b783be","schema_version":"1.0","event_id":"sha256:89cb5e31b6873c00285d26b950f77fc182b45ff21bce0bf028d9fd8f55b783be"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:P6S4L3NSOYDTE35L3LM6RHNRTQ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Long-Time Stability Analysis for Stochastic Evolution Equations with Multiplicative Noise","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Abdelaziz Rhandi, Abdellatif Elgrou, Jawad Salhi","submitted_at":"2026-05-19T23:10:23Z","abstract_excerpt":"In this paper, we study the long-time stability behavior of a class of linear stochastic evolution equations in a Hilbert space with multiplicative noise. Explicit sufficient conditions for $p$-th moment and almost sure exponential stability are established, highlighting the interplay between the principal eigenvalue of the governing operator, the drift coefficient, and the noise intensity. The relationship between these two notions of stability is also clarified. Applications to several stochastic partial differential equations are presented. In addition, a fully discrete spectral Galerkin me"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.20553","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/2605.20553/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-21T01:04:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dVRxnO+wgHq4rKdRsPQC8Ltd9pmfWKUk/sTPancvq1OlslveNFR7F5ubzg1+SoKlv3W3T/JgdNFG+4ASNOlxAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-24T20:43:58.926614Z"},"content_sha256":"8a81658deaf0b2e2c15d36c5f9c307a32ac387959f084e7b9f0f9b76dd0e04fa","schema_version":"1.0","event_id":"sha256:8a81658deaf0b2e2c15d36c5f9c307a32ac387959f084e7b9f0f9b76dd0e04fa"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/P6S4L3NSOYDTE35L3LM6RHNRTQ/bundle.json","state_url":"https://pith.science/pith/P6S4L3NSOYDTE35L3LM6RHNRTQ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/P6S4L3NSOYDTE35L3LM6RHNRTQ/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":"2026-05-24T20:43:58Z","links":{"resolver":"https://pith.science/pith/P6S4L3NSOYDTE35L3LM6RHNRTQ","bundle":"https://pith.science/pith/P6S4L3NSOYDTE35L3LM6RHNRTQ/bundle.json","state":"https://pith.science/pith/P6S4L3NSOYDTE35L3LM6RHNRTQ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/P6S4L3NSOYDTE35L3LM6RHNRTQ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:P6S4L3NSOYDTE35L3LM6RHNRTQ","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":"d6288de461aa746d4a70e09b6d0665237cc8ba6e58375f11c07e74174f1e6238","cross_cats_sorted":[],"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.AP","submitted_at":"2026-05-19T23:10:23Z","title_canon_sha256":"965d3e65a5d042116c15da2ad6887cf849419ff32ba97c0df6b19db8a66a1037"},"schema_version":"1.0","source":{"id":"2605.20553","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.20553","created_at":"2026-05-21T01:04:42Z"},{"alias_kind":"arxiv_version","alias_value":"2605.20553v1","created_at":"2026-05-21T01:04:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.20553","created_at":"2026-05-21T01:04:42Z"},{"alias_kind":"pith_short_12","alias_value":"P6S4L3NSOYDT","created_at":"2026-05-21T01:04:42Z"},{"alias_kind":"pith_short_16","alias_value":"P6S4L3NSOYDTE35L","created_at":"2026-05-21T01:04:42Z"},{"alias_kind":"pith_short_8","alias_value":"P6S4L3NS","created_at":"2026-05-21T01:04:42Z"}],"graph_snapshots":[{"event_id":"sha256:8a81658deaf0b2e2c15d36c5f9c307a32ac387959f084e7b9f0f9b76dd0e04fa","target":"graph","created_at":"2026-05-21T01:04:42Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2605.20553/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper, we study the long-time stability behavior of a class of linear stochastic evolution equations in a Hilbert space with multiplicative noise. Explicit sufficient conditions for $p$-th moment and almost sure exponential stability are established, highlighting the interplay between the principal eigenvalue of the governing operator, the drift coefficient, and the noise intensity. The relationship between these two notions of stability is also clarified. Applications to several stochastic partial differential equations are presented. In addition, a fully discrete spectral Galerkin me","authors_text":"Abdelaziz Rhandi, Abdellatif Elgrou, Jawad Salhi","cross_cats":[],"headline":"","license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.AP","submitted_at":"2026-05-19T23:10:23Z","title":"Long-Time Stability Analysis for Stochastic Evolution Equations with Multiplicative Noise"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.20553","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":"sha256:89cb5e31b6873c00285d26b950f77fc182b45ff21bce0bf028d9fd8f55b783be","target":"record","created_at":"2026-05-21T01:04:42Z","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":"d6288de461aa746d4a70e09b6d0665237cc8ba6e58375f11c07e74174f1e6238","cross_cats_sorted":[],"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.AP","submitted_at":"2026-05-19T23:10:23Z","title_canon_sha256":"965d3e65a5d042116c15da2ad6887cf849419ff32ba97c0df6b19db8a66a1037"},"schema_version":"1.0","source":{"id":"2605.20553","kind":"arxiv","version":1}},"canonical_sha256":"7fa5c5edb27607326fabdad9e89db19c3ef82d65c1e153bc71d0703a45f64b54","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7fa5c5edb27607326fabdad9e89db19c3ef82d65c1e153bc71d0703a45f64b54","first_computed_at":"2026-05-21T01:04:42.269746Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-21T01:04:42.269746Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ZF049oiytufpFHLzjDN1/+GOl1D11DMyc1IrfvGRlEYNijOiw2NKqFPxEvsQA0VOAv76StpIFlMjfvQamq9DBg==","signature_status":"signed_v1","signed_at":"2026-05-21T01:04:42.270446Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.20553","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:89cb5e31b6873c00285d26b950f77fc182b45ff21bce0bf028d9fd8f55b783be","sha256:8a81658deaf0b2e2c15d36c5f9c307a32ac387959f084e7b9f0f9b76dd0e04fa"],"state_sha256":"05d61a44adc50fe098177e53895f16a67139cf53e698e3af929a8b409eac5fb0"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6MZmChTp82Xo2sz3NUeq+Ys/2+S2wJ2E2D1+9fB3d4V5EWcImwOmDIV94x7YNnsprkUaUkOxbC9H9XYbVYgpAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-24T20:43:58.930294Z","bundle_sha256":"df542a01148c41f329e965855b6ffa81cbd20fef92ae94e76da7b5060e0ed299"}}