{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:ZFHV4KN3J6ZQQ7AUHI2QOMF2PC","short_pith_number":"pith:ZFHV4KN3","canonical_record":{"source":{"id":"1803.02973","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-03-08T05:26:29Z","cross_cats_sorted":[],"title_canon_sha256":"e3a8095385b4f481238e9b8fbf983a8fe24d0ffd8f6603f1243cf5788cdb276c","abstract_canon_sha256":"25e69c6d4862ff891e32aa41b000e28d5483e4a5d1d636d10ce815623412f0e8"},"schema_version":"1.0"},"canonical_sha256":"c94f5e29bb4fb3087c143a350730ba78a7e32795e80411f2317f9b9bc1dda00b","source":{"kind":"arxiv","id":"1803.02973","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1803.02973","created_at":"2026-05-18T00:06:56Z"},{"alias_kind":"arxiv_version","alias_value":"1803.02973v2","created_at":"2026-05-18T00:06:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1803.02973","created_at":"2026-05-18T00:06:56Z"},{"alias_kind":"pith_short_12","alias_value":"ZFHV4KN3J6ZQ","created_at":"2026-05-18T12:33:07Z"},{"alias_kind":"pith_short_16","alias_value":"ZFHV4KN3J6ZQQ7AU","created_at":"2026-05-18T12:33:07Z"},{"alias_kind":"pith_short_8","alias_value":"ZFHV4KN3","created_at":"2026-05-18T12:33:07Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:ZFHV4KN3J6ZQQ7AUHI2QOMF2PC","target":"record","payload":{"canonical_record":{"source":{"id":"1803.02973","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-03-08T05:26:29Z","cross_cats_sorted":[],"title_canon_sha256":"e3a8095385b4f481238e9b8fbf983a8fe24d0ffd8f6603f1243cf5788cdb276c","abstract_canon_sha256":"25e69c6d4862ff891e32aa41b000e28d5483e4a5d1d636d10ce815623412f0e8"},"schema_version":"1.0"},"canonical_sha256":"c94f5e29bb4fb3087c143a350730ba78a7e32795e80411f2317f9b9bc1dda00b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:06:56.238279Z","signature_b64":"uW024/65sTSih61o2UfYjXCL+6ZEGQQGeCMSbACKbwoeGpCsJoPnu05OtOqmAX4rV7+f3/jNvxVQFO8LJUK7DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c94f5e29bb4fb3087c143a350730ba78a7e32795e80411f2317f9b9bc1dda00b","last_reissued_at":"2026-05-18T00:06:56.237522Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:06:56.237522Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1803.02973","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:06:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jgIpPZPGraZ0CQn9C4ZMGwqulABdb6xG5ly709GuWvRMrbfsM+cyblbBqAvLppt4Rs0YIj2UtJumR6wS4owfCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T03:13:10.422365Z"},"content_sha256":"5839ae16bb24ff05aa1b86b3d4b51d192bb73aed2b76cdf0108a5b9f963caf43","schema_version":"1.0","event_id":"sha256:5839ae16bb24ff05aa1b86b3d4b51d192bb73aed2b76cdf0108a5b9f963caf43"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:ZFHV4KN3J6ZQQ7AUHI2QOMF2PC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On properties of a class of strong limits for supercritical superprocesses","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Renming Song, Rui Zhang, Yan-Xia Ren","submitted_at":"2018-03-08T05:26:29Z","abstract_excerpt":"Suppose that $X=\\{X_t, t\\ge 0; \\mathbb{P}_{\\mu}\\}$ is a supercritical superprocess in a locally compact separable metric space $E$. Let $\\phi_0$ be a positive\n  eigenfunction corresponding to the first eigenvalue $\\lambda_0$ of the generator of the mean semigroup of $X$. Then $M_t:=e^{-\\lambda_0t}\\langle\\phi_0, X_t\\rangle$ is a positive martingale. Let $M_\\infty$ be the limit of $M_t$. It is known that $M_\\infty$ is non-degenerate iff the $L\\log L$ condition is satisfied. When the $L\\log L$ condition may not be satisfied, we recently proved in (arXiv:1708.04422) that there exist a non-negative"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.02973","kind":"arxiv","version":2},"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":"2026-05-18T00:06:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GXwHb4q8pbM7GryIpXaSFmuMa5AH9lapmcoOYVjzkdbjhrNuJkvYVMqtr8Occo3+l+cPNymy+YVvfrid3Jy3Dg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T03:13:10.422734Z"},"content_sha256":"9cd25822f5cd462eecbcbb4e57f701d26d8ae023d5c82692c99ca82abfa71bae","schema_version":"1.0","event_id":"sha256:9cd25822f5cd462eecbcbb4e57f701d26d8ae023d5c82692c99ca82abfa71bae"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ZFHV4KN3J6ZQQ7AUHI2QOMF2PC/bundle.json","state_url":"https://pith.science/pith/ZFHV4KN3J6ZQQ7AUHI2QOMF2PC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ZFHV4KN3J6ZQQ7AUHI2QOMF2PC/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":"2026-06-01T03:13:10Z","links":{"resolver":"https://pith.science/pith/ZFHV4KN3J6ZQQ7AUHI2QOMF2PC","bundle":"https://pith.science/pith/ZFHV4KN3J6ZQQ7AUHI2QOMF2PC/bundle.json","state":"https://pith.science/pith/ZFHV4KN3J6ZQQ7AUHI2QOMF2PC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ZFHV4KN3J6ZQQ7AUHI2QOMF2PC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:ZFHV4KN3J6ZQQ7AUHI2QOMF2PC","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":"25e69c6d4862ff891e32aa41b000e28d5483e4a5d1d636d10ce815623412f0e8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-03-08T05:26:29Z","title_canon_sha256":"e3a8095385b4f481238e9b8fbf983a8fe24d0ffd8f6603f1243cf5788cdb276c"},"schema_version":"1.0","source":{"id":"1803.02973","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1803.02973","created_at":"2026-05-18T00:06:56Z"},{"alias_kind":"arxiv_version","alias_value":"1803.02973v2","created_at":"2026-05-18T00:06:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1803.02973","created_at":"2026-05-18T00:06:56Z"},{"alias_kind":"pith_short_12","alias_value":"ZFHV4KN3J6ZQ","created_at":"2026-05-18T12:33:07Z"},{"alias_kind":"pith_short_16","alias_value":"ZFHV4KN3J6ZQQ7AU","created_at":"2026-05-18T12:33:07Z"},{"alias_kind":"pith_short_8","alias_value":"ZFHV4KN3","created_at":"2026-05-18T12:33:07Z"}],"graph_snapshots":[{"event_id":"sha256:9cd25822f5cd462eecbcbb4e57f701d26d8ae023d5c82692c99ca82abfa71bae","target":"graph","created_at":"2026-05-18T00:06:56Z","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":"Suppose that $X=\\{X_t, t\\ge 0; \\mathbb{P}_{\\mu}\\}$ is a supercritical superprocess in a locally compact separable metric space $E$. Let $\\phi_0$ be a positive\n  eigenfunction corresponding to the first eigenvalue $\\lambda_0$ of the generator of the mean semigroup of $X$. Then $M_t:=e^{-\\lambda_0t}\\langle\\phi_0, X_t\\rangle$ is a positive martingale. Let $M_\\infty$ be the limit of $M_t$. It is known that $M_\\infty$ is non-degenerate iff the $L\\log L$ condition is satisfied. When the $L\\log L$ condition may not be satisfied, we recently proved in (arXiv:1708.04422) that there exist a non-negative","authors_text":"Renming Song, Rui Zhang, Yan-Xia Ren","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-03-08T05:26:29Z","title":"On properties of a class of strong limits for supercritical superprocesses"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.02973","kind":"arxiv","version":2},"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":"sha256:5839ae16bb24ff05aa1b86b3d4b51d192bb73aed2b76cdf0108a5b9f963caf43","target":"record","created_at":"2026-05-18T00:06:56Z","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":"25e69c6d4862ff891e32aa41b000e28d5483e4a5d1d636d10ce815623412f0e8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-03-08T05:26:29Z","title_canon_sha256":"e3a8095385b4f481238e9b8fbf983a8fe24d0ffd8f6603f1243cf5788cdb276c"},"schema_version":"1.0","source":{"id":"1803.02973","kind":"arxiv","version":2}},"canonical_sha256":"c94f5e29bb4fb3087c143a350730ba78a7e32795e80411f2317f9b9bc1dda00b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c94f5e29bb4fb3087c143a350730ba78a7e32795e80411f2317f9b9bc1dda00b","first_computed_at":"2026-05-18T00:06:56.237522Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:06:56.237522Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"uW024/65sTSih61o2UfYjXCL+6ZEGQQGeCMSbACKbwoeGpCsJoPnu05OtOqmAX4rV7+f3/jNvxVQFO8LJUK7DA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:06:56.238279Z","signed_message":"canonical_sha256_bytes"},"source_id":"1803.02973","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5839ae16bb24ff05aa1b86b3d4b51d192bb73aed2b76cdf0108a5b9f963caf43","sha256:9cd25822f5cd462eecbcbb4e57f701d26d8ae023d5c82692c99ca82abfa71bae"],"state_sha256":"148d2f915dcbad5d1e4b8cf24798efe6ca9aabd520306476e3a042f7feabd73a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"UsDVORDlF9vo2KAuMGu9LMYyLqCZK3WUO/HkgjgF+qIFG1ZyYukPpJFmcrJ0/x8wQ6F90WmLLErgyZ19QxckBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T03:13:10.424887Z","bundle_sha256":"9accdfbf7b67b74acb19ad3a24625caa827cd261adb019948db8ca8e595cac1a"}}