{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:F2CQ4K3SEMO3IST7ZMMOB674WW","short_pith_number":"pith:F2CQ4K3S","canonical_record":{"source":{"id":"1703.00330","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-03-01T15:05:29Z","cross_cats_sorted":[],"title_canon_sha256":"0275e9e50be00200a5609ef72f2e540a517f6a3f8ecf5b267589a69e3f9fbf8f","abstract_canon_sha256":"830522ebfaa0c721a48f97210561cad9cd00998fc52b45ae5be465380e513bbc"},"schema_version":"1.0"},"canonical_sha256":"2e850e2b72231db44a7fcb18e0fbfcb5a0302ffdd6a0eaf7e1627c8297dacc9b","source":{"kind":"arxiv","id":"1703.00330","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1703.00330","created_at":"2026-05-18T00:49:43Z"},{"alias_kind":"arxiv_version","alias_value":"1703.00330v1","created_at":"2026-05-18T00:49:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1703.00330","created_at":"2026-05-18T00:49:43Z"},{"alias_kind":"pith_short_12","alias_value":"F2CQ4K3SEMO3","created_at":"2026-05-18T12:31:12Z"},{"alias_kind":"pith_short_16","alias_value":"F2CQ4K3SEMO3IST7","created_at":"2026-05-18T12:31:12Z"},{"alias_kind":"pith_short_8","alias_value":"F2CQ4K3S","created_at":"2026-05-18T12:31:12Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:F2CQ4K3SEMO3IST7ZMMOB674WW","target":"record","payload":{"canonical_record":{"source":{"id":"1703.00330","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-03-01T15:05:29Z","cross_cats_sorted":[],"title_canon_sha256":"0275e9e50be00200a5609ef72f2e540a517f6a3f8ecf5b267589a69e3f9fbf8f","abstract_canon_sha256":"830522ebfaa0c721a48f97210561cad9cd00998fc52b45ae5be465380e513bbc"},"schema_version":"1.0"},"canonical_sha256":"2e850e2b72231db44a7fcb18e0fbfcb5a0302ffdd6a0eaf7e1627c8297dacc9b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:49:43.219374Z","signature_b64":"jH/vHwPD+d6zgweKLMWgw89yh2pgIqJCzmyQptCOghriFC6SEdmowh6q9EPYkjUEsNgG/ltUbekTwji9NspJBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2e850e2b72231db44a7fcb18e0fbfcb5a0302ffdd6a0eaf7e1627c8297dacc9b","last_reissued_at":"2026-05-18T00:49:43.218759Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:49:43.218759Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1703.00330","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":"2026-05-18T00:49:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gPpB7ZboYAybejHVb3BQb8Z1qvvyIe6ogrtIhirPCQj8h//iHRucwgkYMin9cHL7vrn7F5NOZRJJ2OljpnNKBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T13:13:40.192728Z"},"content_sha256":"0a2b930505875c488729e84513a247ff303832d41e7c522ec08869266c56815a","schema_version":"1.0","event_id":"sha256:0a2b930505875c488729e84513a247ff303832d41e7c522ec08869266c56815a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:F2CQ4K3SEMO3IST7ZMMOB674WW","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Convolution Semigroups of Probability Measures on Gelfand Pairs, Revisited","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"David Applebaum","submitted_at":"2017-03-01T15:05:29Z","abstract_excerpt":"Our goal is to find classes of convolution semigroups on Lie groups $G$ that give rise to interesting processes in symmetric spaces $G/K$. The $K$-bi-invariant convolution semigroups are a well-studied example. An appealing direction for the next step is to generalise to right $K$-invariant convolution semigroups, but recent work of Liao has shown that these are in one-to-one correspondence with $K$-bi-invariant convolution semigroups. We investigate a weaker notion of right $K$-invariance, but show that this is, in fact, the same as the usual notion. Another possible approach is to use genera"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.00330","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":"2026-05-18T00:49:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/sea0pL/b+3dk7qvJCto8nf7Zl0/1qsmlT+uTwGufAO4Cq2HTeASJZ4gwHImbEiIyhqAquJJWuA04u2WEiM6DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T13:13:40.193069Z"},"content_sha256":"f768209e8d282d20466b3cbc2204bb359efedce291b0b1cd67a3b9ef90e7bbdb","schema_version":"1.0","event_id":"sha256:f768209e8d282d20466b3cbc2204bb359efedce291b0b1cd67a3b9ef90e7bbdb"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/F2CQ4K3SEMO3IST7ZMMOB674WW/bundle.json","state_url":"https://pith.science/pith/F2CQ4K3SEMO3IST7ZMMOB674WW/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/F2CQ4K3SEMO3IST7ZMMOB674WW/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":"2026-06-27T13:13:40Z","links":{"resolver":"https://pith.science/pith/F2CQ4K3SEMO3IST7ZMMOB674WW","bundle":"https://pith.science/pith/F2CQ4K3SEMO3IST7ZMMOB674WW/bundle.json","state":"https://pith.science/pith/F2CQ4K3SEMO3IST7ZMMOB674WW/state.json","well_known_bundle":"https://pith.science/.well-known/pith/F2CQ4K3SEMO3IST7ZMMOB674WW/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:F2CQ4K3SEMO3IST7ZMMOB674WW","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":"830522ebfaa0c721a48f97210561cad9cd00998fc52b45ae5be465380e513bbc","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-03-01T15:05:29Z","title_canon_sha256":"0275e9e50be00200a5609ef72f2e540a517f6a3f8ecf5b267589a69e3f9fbf8f"},"schema_version":"1.0","source":{"id":"1703.00330","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1703.00330","created_at":"2026-05-18T00:49:43Z"},{"alias_kind":"arxiv_version","alias_value":"1703.00330v1","created_at":"2026-05-18T00:49:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1703.00330","created_at":"2026-05-18T00:49:43Z"},{"alias_kind":"pith_short_12","alias_value":"F2CQ4K3SEMO3","created_at":"2026-05-18T12:31:12Z"},{"alias_kind":"pith_short_16","alias_value":"F2CQ4K3SEMO3IST7","created_at":"2026-05-18T12:31:12Z"},{"alias_kind":"pith_short_8","alias_value":"F2CQ4K3S","created_at":"2026-05-18T12:31:12Z"}],"graph_snapshots":[{"event_id":"sha256:f768209e8d282d20466b3cbc2204bb359efedce291b0b1cd67a3b9ef90e7bbdb","target":"graph","created_at":"2026-05-18T00:49:43Z","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":"Our goal is to find classes of convolution semigroups on Lie groups $G$ that give rise to interesting processes in symmetric spaces $G/K$. The $K$-bi-invariant convolution semigroups are a well-studied example. An appealing direction for the next step is to generalise to right $K$-invariant convolution semigroups, but recent work of Liao has shown that these are in one-to-one correspondence with $K$-bi-invariant convolution semigroups. We investigate a weaker notion of right $K$-invariance, but show that this is, in fact, the same as the usual notion. Another possible approach is to use genera","authors_text":"David Applebaum","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-03-01T15:05:29Z","title":"Convolution Semigroups of Probability Measures on Gelfand Pairs, Revisited"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.00330","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":"sha256:0a2b930505875c488729e84513a247ff303832d41e7c522ec08869266c56815a","target":"record","created_at":"2026-05-18T00:49:43Z","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":"830522ebfaa0c721a48f97210561cad9cd00998fc52b45ae5be465380e513bbc","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-03-01T15:05:29Z","title_canon_sha256":"0275e9e50be00200a5609ef72f2e540a517f6a3f8ecf5b267589a69e3f9fbf8f"},"schema_version":"1.0","source":{"id":"1703.00330","kind":"arxiv","version":1}},"canonical_sha256":"2e850e2b72231db44a7fcb18e0fbfcb5a0302ffdd6a0eaf7e1627c8297dacc9b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2e850e2b72231db44a7fcb18e0fbfcb5a0302ffdd6a0eaf7e1627c8297dacc9b","first_computed_at":"2026-05-18T00:49:43.218759Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:49:43.218759Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"jH/vHwPD+d6zgweKLMWgw89yh2pgIqJCzmyQptCOghriFC6SEdmowh6q9EPYkjUEsNgG/ltUbekTwji9NspJBw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:49:43.219374Z","signed_message":"canonical_sha256_bytes"},"source_id":"1703.00330","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0a2b930505875c488729e84513a247ff303832d41e7c522ec08869266c56815a","sha256:f768209e8d282d20466b3cbc2204bb359efedce291b0b1cd67a3b9ef90e7bbdb"],"state_sha256":"b0bf3c406c9fd57f61bc245066e0de8d714eaa6a43b77a7aac1f4519becb21fd"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/0Ki+HFs2nvPW5Acwg1n0HOfmQhrJkc0JWZ936ONa+juajL2Ftp0tkaCQJ7AyJKTgzSNu/teQb3ghfW+jPhcBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-27T13:13:40.195091Z","bundle_sha256":"b4e398e6881435fa9a9b1f7dbe087950e3f4ba1de6e42f8546e0b3162a1ce4a6"}}