{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:UYENY4LKSV62X4XRHGYFX7OVSA","short_pith_number":"pith:UYENY4LK","canonical_record":{"source":{"id":"1207.0202","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-07-01T12:24:26Z","cross_cats_sorted":[],"title_canon_sha256":"cdf59cc5b59076e11bf1da66f6d2318a65d4d1155f2054e2cb5d0b86e7fb1d48","abstract_canon_sha256":"ac72facd0dcced186641a006a99514bbb64b1871a67ad1e3c4225b3f5228759d"},"schema_version":"1.0"},"canonical_sha256":"a608dc716a957dabf2f139b05bfdd5902ae74a9f69cf637e044447a0c7a5d9f0","source":{"kind":"arxiv","id":"1207.0202","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1207.0202","created_at":"2026-05-18T03:51:59Z"},{"alias_kind":"arxiv_version","alias_value":"1207.0202v1","created_at":"2026-05-18T03:51:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.0202","created_at":"2026-05-18T03:51:59Z"},{"alias_kind":"pith_short_12","alias_value":"UYENY4LKSV62","created_at":"2026-05-18T12:27:25Z"},{"alias_kind":"pith_short_16","alias_value":"UYENY4LKSV62X4XR","created_at":"2026-05-18T12:27:25Z"},{"alias_kind":"pith_short_8","alias_value":"UYENY4LK","created_at":"2026-05-18T12:27:25Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:UYENY4LKSV62X4XRHGYFX7OVSA","target":"record","payload":{"canonical_record":{"source":{"id":"1207.0202","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-07-01T12:24:26Z","cross_cats_sorted":[],"title_canon_sha256":"cdf59cc5b59076e11bf1da66f6d2318a65d4d1155f2054e2cb5d0b86e7fb1d48","abstract_canon_sha256":"ac72facd0dcced186641a006a99514bbb64b1871a67ad1e3c4225b3f5228759d"},"schema_version":"1.0"},"canonical_sha256":"a608dc716a957dabf2f139b05bfdd5902ae74a9f69cf637e044447a0c7a5d9f0","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:51:59.898298Z","signature_b64":"Aoc5fiN6OP7ctj2yvr4S0fgvr5Q5GtX06B+nRiDZKFq0RboAoLOB47k5Xs/DAjiwqpfqJHPem1lUoPHbux9VDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a608dc716a957dabf2f139b05bfdd5902ae74a9f69cf637e044447a0c7a5d9f0","last_reissued_at":"2026-05-18T03:51:59.897881Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:51:59.897881Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1207.0202","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":"2026-05-18T03:51:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zMM25s4EGGXYikNVr1+7DGlRXBfGg/BKx9zA7wu4v8xMkW4LmMHPq2p2zI7fsPyxV9OeXJsPUHwLza0lE6pqBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T22:39:52.744979Z"},"content_sha256":"b34c7a373efd90dad776c1815b38ee57cadfd82d22a592a8f4793e25fe17710a","schema_version":"1.0","event_id":"sha256:b34c7a373efd90dad776c1815b38ee57cadfd82d22a592a8f4793e25fe17710a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:UYENY4LKSV62X4XRHGYFX7OVSA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Branching diffusion in the super-critical regime","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Leonid Koralov, Stanislav Molchanov","submitted_at":"2012-07-01T12:24:26Z","abstract_excerpt":"We investigate the long-time evolution of branching diffusion processes (starting with a single particle) in inhomogeneous media. The qualitative behavior of the processes depends on the intensity of the branching. We analyze the super-critical case, when the total number of particles growing exponentially with positive probability. We study the asymptotics of the number of particles in different regions of space and describe the growth of the region occupied by the particles."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.0202","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":"2026-05-18T03:51:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"duRQ8bijwcWvEqsGBlurqJN/MLykVgAV99oGgu+2PYz/TPvMv3STPtUR2SBfPrBwX9THuJ7DrgANQYGPLNnzAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T22:39:52.745336Z"},"content_sha256":"5e9e02ab8b075b34742cb353c606aba9ddce25b52fca0d9f6e191532a5274a86","schema_version":"1.0","event_id":"sha256:5e9e02ab8b075b34742cb353c606aba9ddce25b52fca0d9f6e191532a5274a86"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/UYENY4LKSV62X4XRHGYFX7OVSA/bundle.json","state_url":"https://pith.science/pith/UYENY4LKSV62X4XRHGYFX7OVSA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/UYENY4LKSV62X4XRHGYFX7OVSA/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":"2026-06-24T22:39:52Z","links":{"resolver":"https://pith.science/pith/UYENY4LKSV62X4XRHGYFX7OVSA","bundle":"https://pith.science/pith/UYENY4LKSV62X4XRHGYFX7OVSA/bundle.json","state":"https://pith.science/pith/UYENY4LKSV62X4XRHGYFX7OVSA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/UYENY4LKSV62X4XRHGYFX7OVSA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:UYENY4LKSV62X4XRHGYFX7OVSA","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":"ac72facd0dcced186641a006a99514bbb64b1871a67ad1e3c4225b3f5228759d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-07-01T12:24:26Z","title_canon_sha256":"cdf59cc5b59076e11bf1da66f6d2318a65d4d1155f2054e2cb5d0b86e7fb1d48"},"schema_version":"1.0","source":{"id":"1207.0202","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1207.0202","created_at":"2026-05-18T03:51:59Z"},{"alias_kind":"arxiv_version","alias_value":"1207.0202v1","created_at":"2026-05-18T03:51:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.0202","created_at":"2026-05-18T03:51:59Z"},{"alias_kind":"pith_short_12","alias_value":"UYENY4LKSV62","created_at":"2026-05-18T12:27:25Z"},{"alias_kind":"pith_short_16","alias_value":"UYENY4LKSV62X4XR","created_at":"2026-05-18T12:27:25Z"},{"alias_kind":"pith_short_8","alias_value":"UYENY4LK","created_at":"2026-05-18T12:27:25Z"}],"graph_snapshots":[{"event_id":"sha256:5e9e02ab8b075b34742cb353c606aba9ddce25b52fca0d9f6e191532a5274a86","target":"graph","created_at":"2026-05-18T03:51:59Z","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 investigate the long-time evolution of branching diffusion processes (starting with a single particle) in inhomogeneous media. The qualitative behavior of the processes depends on the intensity of the branching. We analyze the super-critical case, when the total number of particles growing exponentially with positive probability. We study the asymptotics of the number of particles in different regions of space and describe the growth of the region occupied by the particles.","authors_text":"Leonid Koralov, Stanislav Molchanov","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-07-01T12:24:26Z","title":"Branching diffusion in the super-critical regime"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.0202","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:b34c7a373efd90dad776c1815b38ee57cadfd82d22a592a8f4793e25fe17710a","target":"record","created_at":"2026-05-18T03:51:59Z","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":"ac72facd0dcced186641a006a99514bbb64b1871a67ad1e3c4225b3f5228759d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-07-01T12:24:26Z","title_canon_sha256":"cdf59cc5b59076e11bf1da66f6d2318a65d4d1155f2054e2cb5d0b86e7fb1d48"},"schema_version":"1.0","source":{"id":"1207.0202","kind":"arxiv","version":1}},"canonical_sha256":"a608dc716a957dabf2f139b05bfdd5902ae74a9f69cf637e044447a0c7a5d9f0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a608dc716a957dabf2f139b05bfdd5902ae74a9f69cf637e044447a0c7a5d9f0","first_computed_at":"2026-05-18T03:51:59.897881Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:51:59.897881Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Aoc5fiN6OP7ctj2yvr4S0fgvr5Q5GtX06B+nRiDZKFq0RboAoLOB47k5Xs/DAjiwqpfqJHPem1lUoPHbux9VDA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:51:59.898298Z","signed_message":"canonical_sha256_bytes"},"source_id":"1207.0202","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b34c7a373efd90dad776c1815b38ee57cadfd82d22a592a8f4793e25fe17710a","sha256:5e9e02ab8b075b34742cb353c606aba9ddce25b52fca0d9f6e191532a5274a86"],"state_sha256":"3fefc71a5ea1dd202f4b3ad967adf450f71519097ffca77be4fe62e6e9abf25d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ny5vFlJo+dX2nxBQLCm2Tu+mj4pFfSXJrwjE4NrsdUswsvpEG8ShFv+RiGO5ZWMeZ5+HlZXl3wmQ6QROn9Z4BQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-24T22:39:52.747626Z","bundle_sha256":"8b46402e2fa409cddaebd46275c7c87ed7824937b63c22aceda8990b9869c1db"}}