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The operator $\\partial^\\beta_t$ is the Caputo fractional derivative while $-(-\\Delta)^{\\alpha/2} $ is the generator of an isotropic stable process and $I^{1-\\beta}_t$ is the fractional integral operator. The forcing noise denoted by $\\stackrel{\\cdot}{F}(t,x)$ is a Gaussian noise. And the multiplicative non-linearity $\\s"},"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":"1505.04615","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-05-18T12:47:28Z","cross_cats_sorted":[],"title_canon_sha256":"acacfc12fde62e67dd98a7a332646da9fa36956bc588dc4baf13d6252c804068","abstract_canon_sha256":"4f9cb4db770fdb09b9016a198188bf868b4fb1ebc9d1164979f4a1ffd91b1728"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:07:25.009994Z","signature_b64":"TtXnjeU3//tbjwR6/2IV7Uqqb9Ny3CXXkkZh53uxwZqTTJzzNaNVTUHQN1ZXTA6AcMaoT17IHjnccqgOS9ZwBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"50c21cf0f292c4272e2e1942f82d89116a54390e8d1775fe93863d44ba09d50e","last_reissued_at":"2026-05-18T02:07:25.009563Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:07:25.009563Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Asymptotic properties of some space-time fractional stochastic equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Erkan Nane, Mohammud Foondun","submitted_at":"2015-05-18T12:47:28Z","abstract_excerpt":"Consider non-linear time-fractional stochastic heat type equations of the following type, $$\\partial^\\beta_tu_t(x)=-\\nu(-\\Delta)^{\\alpha/2} u_t(x)+I^{1-\\beta}_t[\\lambda \\sigma(u)\\stackrel{\\cdot}{F}(t,x)]$$ in $(d+1)$ dimensions, where $\\nu>0, \\beta\\in (0,1)$, $\\alpha\\in (0,2]$. The operator $\\partial^\\beta_t$ is the Caputo fractional derivative while $-(-\\Delta)^{\\alpha/2} $ is the generator of an isotropic stable process and $I^{1-\\beta}_t$ is the fractional integral operator. The forcing noise denoted by $\\stackrel{\\cdot}{F}(t,x)$ is a Gaussian noise. 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