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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"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"},"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"},"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 $X^{(\\alpha)}$. 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