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We show that this stochastic PDE has a random-field solution for every finite initial measure $u_0$. Tight a priori bounds on the moments of the solution are also obtained.\n  In the particular case that $\\mathcal{L}f=cf\"$ for some $c>0$, we prove that if $u_0$ is a finite measure of compact support, then the sol"},"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":"1110.4079","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-10-18T18:34:37Z","cross_cats_sorted":[],"title_canon_sha256":"d67bb74058f76531da55a12d6427175ab5d3ba465849b0c0dcb8803f380357b8","abstract_canon_sha256":"e20572ef1726f4a56185f855f27779554bad37e35e57c5a55b439b5952c49664"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:10:47.395449Z","signature_b64":"IX11W9QPUTRpKU/UJV//cQriJ5jYhBX9WH2qi4nBJ94QzaxqQkuZuzT+L4C30Sn/PLqMqLT3rg+RHSkcg72uBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"aff8515048beed8a5c2a3bb7605bde9c9148eceb56b412b2c231df70a95e1e02","last_reissued_at":"2026-05-18T04:10:47.395010Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:10:47.395010Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Initial measures for the stochastic heat equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Daniel Conus, Davar Khoshnevisan, Mathew Joseph, Shang-Yuan Shiu","submitted_at":"2011-10-18T18:34:37Z","abstract_excerpt":"We consider a family of nonlinear stochastic heat equations of the form $\\partial_t u=\\mathcal{L}u + \\sigma(u)\\dot{W}$, where $\\dot{W}$ denotes space-time white noise, $\\mathcal{L}$ the generator of a symmetric L\\'evy process on $\\R$, and $\\sigma$ is Lipschitz continuous and zero at 0. 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