{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:7POZTZD4WJ7RPH2DOTSHXBBOLU","short_pith_number":"pith:7POZTZD4","canonical_record":{"source":{"id":"1007.2904","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-07-17T07:58:38Z","cross_cats_sorted":[],"title_canon_sha256":"f3e7bfaf190262ae8abf4723bcc4126df989678b43d6334ff265349de7414992","abstract_canon_sha256":"f7cbbf40200cd68dbbf526ea33f51c57ddc5f14a50b128166388277744c90c25"},"schema_version":"1.0"},"canonical_sha256":"fbdd99e47cb27f179f4374e47b842e5d0d3de6aca09ba113dd626bbe871d68df","source":{"kind":"arxiv","id":"1007.2904","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1007.2904","created_at":"2026-05-18T04:32:11Z"},{"alias_kind":"arxiv_version","alias_value":"1007.2904v2","created_at":"2026-05-18T04:32:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1007.2904","created_at":"2026-05-18T04:32:11Z"},{"alias_kind":"pith_short_12","alias_value":"7POZTZD4WJ7R","created_at":"2026-05-18T12:26:05Z"},{"alias_kind":"pith_short_16","alias_value":"7POZTZD4WJ7RPH2D","created_at":"2026-05-18T12:26:05Z"},{"alias_kind":"pith_short_8","alias_value":"7POZTZD4","created_at":"2026-05-18T12:26:05Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:7POZTZD4WJ7RPH2DOTSHXBBOLU","target":"record","payload":{"canonical_record":{"source":{"id":"1007.2904","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-07-17T07:58:38Z","cross_cats_sorted":[],"title_canon_sha256":"f3e7bfaf190262ae8abf4723bcc4126df989678b43d6334ff265349de7414992","abstract_canon_sha256":"f7cbbf40200cd68dbbf526ea33f51c57ddc5f14a50b128166388277744c90c25"},"schema_version":"1.0"},"canonical_sha256":"fbdd99e47cb27f179f4374e47b842e5d0d3de6aca09ba113dd626bbe871d68df","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:32:11.845029Z","signature_b64":"StcdT3fGBwZEHL7jNCNqTeSAKZcAKZZu1jFn/QP5bnvn9DXQc8RYX2Bpi2POHRqtqsYjoFT9MSkwsk8kbzdvBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fbdd99e47cb27f179f4374e47b842e5d0d3de6aca09ba113dd626bbe871d68df","last_reissued_at":"2026-05-18T04:32:11.844521Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:32:11.844521Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1007.2904","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":"2026-05-18T04:32:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GkLZcg12H5zuEm4p3x0Lb9vaM2DabO80QeM/Lamtn7S8gC+hxzoraJS0p4/16hL49PFNgeeqpFs5TfblXhmeCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T15:41:30.739656Z"},"content_sha256":"9faa2cbb50cce38a57bade1b5312c091d4b55f1f920159d53e09a3e1b422388d","schema_version":"1.0","event_id":"sha256:9faa2cbb50cce38a57bade1b5312c091d4b55f1f920159d53e09a3e1b422388d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:7POZTZD4WJ7RPH2DOTSHXBBOLU","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Karhunen-Loeve expansions of alpha-Wiener bridges","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Endre Igloi, Matyas Barczy","submitted_at":"2010-07-17T07:58:38Z","abstract_excerpt":"We study Karhunen-Loeve expansions of the process $(X_t^{(\\alpha)})_{t\\in[0,T)}$ given by the stochastic differential equation $dX_t^{(\\alpha)} = -\\frac\\alpha{T-t} X_t^{(\\alpha)} dt+ dB_t,$ $t\\in[0,T),$ with an initial condition $X_0^{(\\alpha)}=0,$ where $\\alpha>0,$ $T\\in(0,\\infty)$ and $(B_t)_{t\\geq 0}$ is a standard Wiener process. This process is called an $\\alpha$-Wiener bridge or a scaled Brownian bridge, and in the special case of $\\alpha=1$ the usual Wiener bridge. We present weighted and unweighted Karhunen-Loeve expansions of $X^{(\\alpha)}$. As applications, we calculate the Laplace t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1007.2904","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":"2026-05-18T04:32:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VRYsoAKk+JcgBVWY4xuG53sTb21+61BJk+nkUb1nYwqA5ToruM5py8IJWMs7OsTZ5CyJ7/PpZR2AodkDIRlmAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T15:41:30.739994Z"},"content_sha256":"96db4306b0a817c78589f2bf7fb9ed31091e1d2af999aa78d554ca3b118c1d6b","schema_version":"1.0","event_id":"sha256:96db4306b0a817c78589f2bf7fb9ed31091e1d2af999aa78d554ca3b118c1d6b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7POZTZD4WJ7RPH2DOTSHXBBOLU/bundle.json","state_url":"https://pith.science/pith/7POZTZD4WJ7RPH2DOTSHXBBOLU/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7POZTZD4WJ7RPH2DOTSHXBBOLU/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":"2026-06-20T15:41:30Z","links":{"resolver":"https://pith.science/pith/7POZTZD4WJ7RPH2DOTSHXBBOLU","bundle":"https://pith.science/pith/7POZTZD4WJ7RPH2DOTSHXBBOLU/bundle.json","state":"https://pith.science/pith/7POZTZD4WJ7RPH2DOTSHXBBOLU/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7POZTZD4WJ7RPH2DOTSHXBBOLU/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:7POZTZD4WJ7RPH2DOTSHXBBOLU","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":"f7cbbf40200cd68dbbf526ea33f51c57ddc5f14a50b128166388277744c90c25","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-07-17T07:58:38Z","title_canon_sha256":"f3e7bfaf190262ae8abf4723bcc4126df989678b43d6334ff265349de7414992"},"schema_version":"1.0","source":{"id":"1007.2904","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1007.2904","created_at":"2026-05-18T04:32:11Z"},{"alias_kind":"arxiv_version","alias_value":"1007.2904v2","created_at":"2026-05-18T04:32:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1007.2904","created_at":"2026-05-18T04:32:11Z"},{"alias_kind":"pith_short_12","alias_value":"7POZTZD4WJ7R","created_at":"2026-05-18T12:26:05Z"},{"alias_kind":"pith_short_16","alias_value":"7POZTZD4WJ7RPH2D","created_at":"2026-05-18T12:26:05Z"},{"alias_kind":"pith_short_8","alias_value":"7POZTZD4","created_at":"2026-05-18T12:26:05Z"}],"graph_snapshots":[{"event_id":"sha256:96db4306b0a817c78589f2bf7fb9ed31091e1d2af999aa78d554ca3b118c1d6b","target":"graph","created_at":"2026-05-18T04:32:11Z","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 study Karhunen-Loeve expansions of the process $(X_t^{(\\alpha)})_{t\\in[0,T)}$ given by the stochastic differential equation $dX_t^{(\\alpha)} = -\\frac\\alpha{T-t} X_t^{(\\alpha)} dt+ dB_t,$ $t\\in[0,T),$ with an initial condition $X_0^{(\\alpha)}=0,$ where $\\alpha>0,$ $T\\in(0,\\infty)$ and $(B_t)_{t\\geq 0}$ is a standard Wiener process. This process is called an $\\alpha$-Wiener bridge or a scaled Brownian bridge, and in the special case of $\\alpha=1$ the usual Wiener bridge. We present weighted and unweighted Karhunen-Loeve expansions of $X^{(\\alpha)}$. As applications, we calculate the Laplace t","authors_text":"Endre Igloi, Matyas Barczy","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-07-17T07:58:38Z","title":"Karhunen-Loeve expansions of alpha-Wiener bridges"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1007.2904","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":"sha256:9faa2cbb50cce38a57bade1b5312c091d4b55f1f920159d53e09a3e1b422388d","target":"record","created_at":"2026-05-18T04:32:11Z","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":"f7cbbf40200cd68dbbf526ea33f51c57ddc5f14a50b128166388277744c90c25","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-07-17T07:58:38Z","title_canon_sha256":"f3e7bfaf190262ae8abf4723bcc4126df989678b43d6334ff265349de7414992"},"schema_version":"1.0","source":{"id":"1007.2904","kind":"arxiv","version":2}},"canonical_sha256":"fbdd99e47cb27f179f4374e47b842e5d0d3de6aca09ba113dd626bbe871d68df","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fbdd99e47cb27f179f4374e47b842e5d0d3de6aca09ba113dd626bbe871d68df","first_computed_at":"2026-05-18T04:32:11.844521Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:32:11.844521Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"StcdT3fGBwZEHL7jNCNqTeSAKZcAKZZu1jFn/QP5bnvn9DXQc8RYX2Bpi2POHRqtqsYjoFT9MSkwsk8kbzdvBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:32:11.845029Z","signed_message":"canonical_sha256_bytes"},"source_id":"1007.2904","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9faa2cbb50cce38a57bade1b5312c091d4b55f1f920159d53e09a3e1b422388d","sha256:96db4306b0a817c78589f2bf7fb9ed31091e1d2af999aa78d554ca3b118c1d6b"],"state_sha256":"0fc764b7c1605f205a7b77cc90ed9f631748f0867372819726988ad1a0fe1e02"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Jb/eP/fdiF1HckNmLD65usC1oI0nFw19JW9UR/tGdHO8sqzNMAeUgI1R/s1OMe62MrmcmBr7EezM5kPsoR3ICQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-20T15:41:30.741814Z","bundle_sha256":"d4fafeb07b4184a69dba5f7c2a33d620950ea71e5c04f5c07b8450641252b156"}}