{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:DLOTFCNZMBYCZED3LNDGIDH2CA","short_pith_number":"pith:DLOTFCNZ","canonical_record":{"source":{"id":"2607.07410","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2026-07-08T13:42:27Z","cross_cats_sorted":["math.SP"],"title_canon_sha256":"00142e9b5f7c640cdba975de03d197afc64f77841a761adee774b6fccf3bee92","abstract_canon_sha256":"38e4e39f1d6f09c6eef90376f445bd37bc19499a64b93ca7b087c170fd2c87d2"},"schema_version":"1.0"},"canonical_sha256":"1add3289b960702c907b5b46640cfa10206edd78d866cc451f7fc4c148153996","source":{"kind":"arxiv","id":"2607.07410","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.07410","created_at":"2026-07-09T01:20:24Z"},{"alias_kind":"arxiv_version","alias_value":"2607.07410v1","created_at":"2026-07-09T01:20:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.07410","created_at":"2026-07-09T01:20:24Z"},{"alias_kind":"pith_short_12","alias_value":"DLOTFCNZMBYC","created_at":"2026-07-09T01:20:24Z"},{"alias_kind":"pith_short_16","alias_value":"DLOTFCNZMBYCZED3","created_at":"2026-07-09T01:20:24Z"},{"alias_kind":"pith_short_8","alias_value":"DLOTFCNZ","created_at":"2026-07-09T01:20:24Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:DLOTFCNZMBYCZED3LNDGIDH2CA","target":"record","payload":{"canonical_record":{"source":{"id":"2607.07410","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2026-07-08T13:42:27Z","cross_cats_sorted":["math.SP"],"title_canon_sha256":"00142e9b5f7c640cdba975de03d197afc64f77841a761adee774b6fccf3bee92","abstract_canon_sha256":"38e4e39f1d6f09c6eef90376f445bd37bc19499a64b93ca7b087c170fd2c87d2"},"schema_version":"1.0"},"canonical_sha256":"1add3289b960702c907b5b46640cfa10206edd78d866cc451f7fc4c148153996","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-09T01:20:24.225971Z","signature_b64":"7f9POjF9jrhEzHeF4t/8vVgnKXLet7CT7M9r9YFLU2mmOPgz/xfnLTPEQ4ZqMB9AQjIC11KB466JFzD1wh1DCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1add3289b960702c907b5b46640cfa10206edd78d866cc451f7fc4c148153996","last_reissued_at":"2026-07-09T01:20:24.225518Z","signature_status":"signed_v1","first_computed_at":"2026-07-09T01:20:24.225518Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2607.07410","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":"2026-07-09T01:20:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"J6Wm4bv69iooClIyGNtQzHGaI3FDYKHaeivVucH1c+CxcfPw4tSFfCYH1aQj8TXubTT0wULvOgK681NJhCCwBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-05T10:59:37.161141Z"},"content_sha256":"8af8c446c21f2b206d5dc2839ff65105f88af847c1e5890a05dfd6cf8d8f5162","schema_version":"1.0","event_id":"sha256:8af8c446c21f2b206d5dc2839ff65105f88af847c1e5890a05dfd6cf8d8f5162"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:DLOTFCNZMBYCZED3LNDGIDH2CA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Essential spectrum for Brox-type diffusion processes","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.SP"],"primary_cat":"math.PR","authors_text":"Jian Wang, Yuichi Shiozawa","submitted_at":"2026-07-08T13:42:27Z","abstract_excerpt":"This paper investigates the essential spectrum and compactness property of Markov semigroups generated by multi-dimensional Brox diffusion processes under two types of random media: stationary Gaussian random fields and semi-selfsimilar L\\'evy random fields. For Gaussian environments satisfying general stationary covariance growth conditions, we prove that the associated semigroup is almost surely noncompact; if the covariance function grows sublinearly at infinity, then the bottom of the essential spectrum vanishes almost surely, with fractional Brownian fields as a concrete example. For mult"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.07410","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/2607.07410/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":"2026-07-09T01:20:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Gkjqt+sGmt7IYfCFVdrTPprrVoxGG70I4hJOp4mkdx9FBOD0jPi0TgbaPMPNTWL8FErtjN55g3o4J48F1km1Bg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-05T10:59:37.161839Z"},"content_sha256":"6db0743328334ac83d116dbb9b102898e907fe135dfe8f5dff2b9359741680ca","schema_version":"1.0","event_id":"sha256:6db0743328334ac83d116dbb9b102898e907fe135dfe8f5dff2b9359741680ca"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/DLOTFCNZMBYCZED3LNDGIDH2CA/bundle.json","state_url":"https://pith.science/pith/DLOTFCNZMBYCZED3LNDGIDH2CA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/DLOTFCNZMBYCZED3LNDGIDH2CA/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":"2026-08-05T10:59:37Z","links":{"resolver":"https://pith.science/pith/DLOTFCNZMBYCZED3LNDGIDH2CA","bundle":"https://pith.science/pith/DLOTFCNZMBYCZED3LNDGIDH2CA/bundle.json","state":"https://pith.science/pith/DLOTFCNZMBYCZED3LNDGIDH2CA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/DLOTFCNZMBYCZED3LNDGIDH2CA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:DLOTFCNZMBYCZED3LNDGIDH2CA","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":"38e4e39f1d6f09c6eef90376f445bd37bc19499a64b93ca7b087c170fd2c87d2","cross_cats_sorted":["math.SP"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2026-07-08T13:42:27Z","title_canon_sha256":"00142e9b5f7c640cdba975de03d197afc64f77841a761adee774b6fccf3bee92"},"schema_version":"1.0","source":{"id":"2607.07410","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.07410","created_at":"2026-07-09T01:20:24Z"},{"alias_kind":"arxiv_version","alias_value":"2607.07410v1","created_at":"2026-07-09T01:20:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.07410","created_at":"2026-07-09T01:20:24Z"},{"alias_kind":"pith_short_12","alias_value":"DLOTFCNZMBYC","created_at":"2026-07-09T01:20:24Z"},{"alias_kind":"pith_short_16","alias_value":"DLOTFCNZMBYCZED3","created_at":"2026-07-09T01:20:24Z"},{"alias_kind":"pith_short_8","alias_value":"DLOTFCNZ","created_at":"2026-07-09T01:20:24Z"}],"graph_snapshots":[{"event_id":"sha256:6db0743328334ac83d116dbb9b102898e907fe135dfe8f5dff2b9359741680ca","target":"graph","created_at":"2026-07-09T01:20:24Z","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/2607.07410/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"This paper investigates the essential spectrum and compactness property of Markov semigroups generated by multi-dimensional Brox diffusion processes under two types of random media: stationary Gaussian random fields and semi-selfsimilar L\\'evy random fields. For Gaussian environments satisfying general stationary covariance growth conditions, we prove that the associated semigroup is almost surely noncompact; if the covariance function grows sublinearly at infinity, then the bottom of the essential spectrum vanishes almost surely, with fractional Brownian fields as a concrete example. For mult","authors_text":"Jian Wang, Yuichi Shiozawa","cross_cats":["math.SP"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2026-07-08T13:42:27Z","title":"Essential spectrum for Brox-type diffusion processes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.07410","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":"sha256:8af8c446c21f2b206d5dc2839ff65105f88af847c1e5890a05dfd6cf8d8f5162","target":"record","created_at":"2026-07-09T01:20:24Z","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":"38e4e39f1d6f09c6eef90376f445bd37bc19499a64b93ca7b087c170fd2c87d2","cross_cats_sorted":["math.SP"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2026-07-08T13:42:27Z","title_canon_sha256":"00142e9b5f7c640cdba975de03d197afc64f77841a761adee774b6fccf3bee92"},"schema_version":"1.0","source":{"id":"2607.07410","kind":"arxiv","version":1}},"canonical_sha256":"1add3289b960702c907b5b46640cfa10206edd78d866cc451f7fc4c148153996","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1add3289b960702c907b5b46640cfa10206edd78d866cc451f7fc4c148153996","first_computed_at":"2026-07-09T01:20:24.225518Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-09T01:20:24.225518Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"7f9POjF9jrhEzHeF4t/8vVgnKXLet7CT7M9r9YFLU2mmOPgz/xfnLTPEQ4ZqMB9AQjIC11KB466JFzD1wh1DCA==","signature_status":"signed_v1","signed_at":"2026-07-09T01:20:24.225971Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.07410","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8af8c446c21f2b206d5dc2839ff65105f88af847c1e5890a05dfd6cf8d8f5162","sha256:6db0743328334ac83d116dbb9b102898e907fe135dfe8f5dff2b9359741680ca"],"state_sha256":"a6c1181263d7de49488fac1d28e47b8b2f35ff6f8520f227f51a4e09e114dd7d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bsdi4LLhUJS8WwUkFKeuPTe3v+m0gLgf0wz/qQu3QIKeMJUF6JmzSf9uZWWLFDFwnGOU37WLX6NFiG8kBBmhDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-05T10:59:37.169610Z","bundle_sha256":"f94837db64ddb887649462a822bb66957be5213da33873c2d0292575028b361e"}}