{"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"}