{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:WNQM5MMMIGT5YKF6GT4CP6WQQR","short_pith_number":"pith:WNQM5MMM","canonical_record":{"source":{"id":"1907.11867","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2019-07-27T07:35:43Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"2efc6b0c9a93077e3ca32de4b27ab50081004a08eba51c96220a7d90fce208f4","abstract_canon_sha256":"b4f516a98caecce47a19323b6cbd0f8717919ff2cdbb7907d787c46616a1610c"},"schema_version":"1.0"},"canonical_sha256":"b360ceb18c41a7dc28be34f827fad08463402469fa3a19d7dcd6f0b5f0e0de2c","source":{"kind":"arxiv","id":"1907.11867","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1907.11867","created_at":"2026-05-17T23:39:22Z"},{"alias_kind":"arxiv_version","alias_value":"1907.11867v1","created_at":"2026-05-17T23:39:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1907.11867","created_at":"2026-05-17T23:39:22Z"},{"alias_kind":"pith_short_12","alias_value":"WNQM5MMMIGT5","created_at":"2026-05-18T12:33:30Z"},{"alias_kind":"pith_short_16","alias_value":"WNQM5MMMIGT5YKF6","created_at":"2026-05-18T12:33:30Z"},{"alias_kind":"pith_short_8","alias_value":"WNQM5MMM","created_at":"2026-05-18T12:33:30Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:WNQM5MMMIGT5YKF6GT4CP6WQQR","target":"record","payload":{"canonical_record":{"source":{"id":"1907.11867","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2019-07-27T07:35:43Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"2efc6b0c9a93077e3ca32de4b27ab50081004a08eba51c96220a7d90fce208f4","abstract_canon_sha256":"b4f516a98caecce47a19323b6cbd0f8717919ff2cdbb7907d787c46616a1610c"},"schema_version":"1.0"},"canonical_sha256":"b360ceb18c41a7dc28be34f827fad08463402469fa3a19d7dcd6f0b5f0e0de2c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:39:22.757995Z","signature_b64":"+dkYSP8I7QFVh/nMQ/7oipYbFQhR7lFVbCWmSouDMDj+v2ZQaz/+7Zpx0ZbEtT/tvtXNS4KFYbqgt2pEkwaqCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b360ceb18c41a7dc28be34f827fad08463402469fa3a19d7dcd6f0b5f0e0de2c","last_reissued_at":"2026-05-17T23:39:22.757270Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:39:22.757270Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1907.11867","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":"2026-05-17T23:39:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RmiV2hGfGLe8SXq2qsGK+6xXrZp/vzEkQ+0ptMh0Fy099aQKbPF192qF1c6hT2gx2dJf8tyHVtmIWm0gY8HiCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T20:35:21.646605Z"},"content_sha256":"55b496240fd48e9d87e923ea0fa53d67fb658095379cb17156d29b33b7de167b","schema_version":"1.0","event_id":"sha256:55b496240fd48e9d87e923ea0fa53d67fb658095379cb17156d29b33b7de167b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:WNQM5MMMIGT5YKF6GT4CP6WQQR","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Maximal inequalities and exponential estimates for stochastic convolutions driven by L\\'{e}vy-type processes in Banach spaces with application to stochastic quasi-geostrophic equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.PR","authors_text":"Jiahui Zhu, Wei Liu, Zdzis{\\l}aw Brze\\'zniak","submitted_at":"2019-07-27T07:35:43Z","abstract_excerpt":"We present remarkably simple proofs of Burkholder-Davis-Gundy inequalities for stochastic integrals and maximal inequalities for stochastic convolutions in Banach spaces driven by L\\'{e}vy-type processes. Exponential estimates for stochastic convolutions are obtained and two versions of It\\^{o}'s formula in Banach spaces are also derived. Based on the obtained maximal inequality, the existence and uniqueness of mild solutions of stochastic quasi-geostrophic equation with L\\'{e}vy noise is established."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.11867","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":"2026-05-17T23:39:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HSSyb4ss2nsOjuCpmg7SLfLm/qNg31YmARJZfqJc2ULdNH5qXWfM4nliGRFhbk/0hX0FIJhjuLbf9JWycK6pDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T20:35:21.646958Z"},"content_sha256":"00deeaba756011ef862d5112e18e91a1ff44b811c981b88137911cf719387608","schema_version":"1.0","event_id":"sha256:00deeaba756011ef862d5112e18e91a1ff44b811c981b88137911cf719387608"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/WNQM5MMMIGT5YKF6GT4CP6WQQR/bundle.json","state_url":"https://pith.science/pith/WNQM5MMMIGT5YKF6GT4CP6WQQR/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/WNQM5MMMIGT5YKF6GT4CP6WQQR/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":"2026-06-10T20:35:21Z","links":{"resolver":"https://pith.science/pith/WNQM5MMMIGT5YKF6GT4CP6WQQR","bundle":"https://pith.science/pith/WNQM5MMMIGT5YKF6GT4CP6WQQR/bundle.json","state":"https://pith.science/pith/WNQM5MMMIGT5YKF6GT4CP6WQQR/state.json","well_known_bundle":"https://pith.science/.well-known/pith/WNQM5MMMIGT5YKF6GT4CP6WQQR/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:WNQM5MMMIGT5YKF6GT4CP6WQQR","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":"b4f516a98caecce47a19323b6cbd0f8717919ff2cdbb7907d787c46616a1610c","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2019-07-27T07:35:43Z","title_canon_sha256":"2efc6b0c9a93077e3ca32de4b27ab50081004a08eba51c96220a7d90fce208f4"},"schema_version":"1.0","source":{"id":"1907.11867","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1907.11867","created_at":"2026-05-17T23:39:22Z"},{"alias_kind":"arxiv_version","alias_value":"1907.11867v1","created_at":"2026-05-17T23:39:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1907.11867","created_at":"2026-05-17T23:39:22Z"},{"alias_kind":"pith_short_12","alias_value":"WNQM5MMMIGT5","created_at":"2026-05-18T12:33:30Z"},{"alias_kind":"pith_short_16","alias_value":"WNQM5MMMIGT5YKF6","created_at":"2026-05-18T12:33:30Z"},{"alias_kind":"pith_short_8","alias_value":"WNQM5MMM","created_at":"2026-05-18T12:33:30Z"}],"graph_snapshots":[{"event_id":"sha256:00deeaba756011ef862d5112e18e91a1ff44b811c981b88137911cf719387608","target":"graph","created_at":"2026-05-17T23:39:22Z","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":"We present remarkably simple proofs of Burkholder-Davis-Gundy inequalities for stochastic integrals and maximal inequalities for stochastic convolutions in Banach spaces driven by L\\'{e}vy-type processes. Exponential estimates for stochastic convolutions are obtained and two versions of It\\^{o}'s formula in Banach spaces are also derived. Based on the obtained maximal inequality, the existence and uniqueness of mild solutions of stochastic quasi-geostrophic equation with L\\'{e}vy noise is established.","authors_text":"Jiahui Zhu, Wei Liu, Zdzis{\\l}aw Brze\\'zniak","cross_cats":["math.AP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2019-07-27T07:35:43Z","title":"Maximal inequalities and exponential estimates for stochastic convolutions driven by L\\'{e}vy-type processes in Banach spaces with application to stochastic quasi-geostrophic equations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.11867","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":"sha256:55b496240fd48e9d87e923ea0fa53d67fb658095379cb17156d29b33b7de167b","target":"record","created_at":"2026-05-17T23:39:22Z","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":"b4f516a98caecce47a19323b6cbd0f8717919ff2cdbb7907d787c46616a1610c","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2019-07-27T07:35:43Z","title_canon_sha256":"2efc6b0c9a93077e3ca32de4b27ab50081004a08eba51c96220a7d90fce208f4"},"schema_version":"1.0","source":{"id":"1907.11867","kind":"arxiv","version":1}},"canonical_sha256":"b360ceb18c41a7dc28be34f827fad08463402469fa3a19d7dcd6f0b5f0e0de2c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b360ceb18c41a7dc28be34f827fad08463402469fa3a19d7dcd6f0b5f0e0de2c","first_computed_at":"2026-05-17T23:39:22.757270Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:39:22.757270Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"+dkYSP8I7QFVh/nMQ/7oipYbFQhR7lFVbCWmSouDMDj+v2ZQaz/+7Zpx0ZbEtT/tvtXNS4KFYbqgt2pEkwaqCg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:39:22.757995Z","signed_message":"canonical_sha256_bytes"},"source_id":"1907.11867","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:55b496240fd48e9d87e923ea0fa53d67fb658095379cb17156d29b33b7de167b","sha256:00deeaba756011ef862d5112e18e91a1ff44b811c981b88137911cf719387608"],"state_sha256":"912e4e0cfe1c3167d71cd8126f9cee40712427ff85fc358071c89a5644882364"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"O680G26vYY1YbJSrDjltoPDHk4VdWg7JDqyQ5MFDTeYm60TJe9yGZL1ET12J+DJhkiu79fiiScIIrr0ffikkBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-10T20:35:21.648845Z","bundle_sha256":"04b706d0012ad8d2c8b1f9d4732d526812efdb5f085adb97b87b8f240e162966"}}