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In this equation, $L$ is a second-order partial differential operator with constant coefficients, $\\sigma$ and $b$ are smooth functions and $\\dot{F}$ is a Gaussian noise, white in time and with a stationary correlation in space. Let $p^\\eps_{t,x}$ denote the density of the law of $u^\\eps(t,x)$ at a fixed point $(t,x)\\in(0,T]\\time"},"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":"1312.1257","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","primary_cat":"math.PR","submitted_at":"2013-12-04T17:40:48Z","cross_cats_sorted":[],"title_canon_sha256":"23d9cabf7359b48d0052c4cbd44ad004f59a85bb50558104d9f2fa3d5338d443","abstract_canon_sha256":"967b303c1c957c327c16dbd14110b0aa0bfc0e27b2f2de4f1be33a999f0722bd"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:20:37.009537Z","signature_b64":"IvdIbK52yweEhvEB1lgXK5lxASV9wb5TJgD1WSR+Ri4DDIA7h5YSsCNMwKW6s9uKvHbpkH4POYlk9Ol3lj4KCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2ecabbbbf16059e2a66265d64821893a6a4e5fa3e258c830cdd782213305e0eb","last_reissued_at":"2026-05-18T02:20:37.008950Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:20:37.008950Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Logarithmic asymptotics of the densities of SPDEs driven by spatially correlated noise","license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Andr\\'e S\\\"u\\ss, Marta Sanz-Sol\\'e","submitted_at":"2013-12-04T17:40:48Z","abstract_excerpt":"We consider the family of stochastic partial differential equations indexed by a parameter $\\eps\\in(0,1]$, \\begin{equation*} Lu^{\\eps}(t,x) = \\eps\\sigma(u^\\eps(t,x))\\dot{F}(t,x)+b(u^\\eps(t,x)), \\end{equation*} $(t,x)\\in(0,T]\\times\\Rd$ with suitable initial conditions. In this equation, $L$ is a second-order partial differential operator with constant coefficients, $\\sigma$ and $b$ are smooth functions and $\\dot{F}$ is a Gaussian noise, white in time and with a stationary correlation in space. 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