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We show that $u$ also satisfies $\\partial_x^2 u +[\\,(-\\partial_t^2)^{1/2}+\\sqrt{2}\\partial_x(-\\partial_t^2)^{1/4}\\,]\\,u^a= \\partial_x\\partial_t{\\tilde B}$ in $R\\times(0,\\infty)$ where $u^a$ stands for the extension of $u(x,t)$ to $(x,t)\\in R^2$ which is antisymmetric in $t$ and $\\tilde{B}$ is another Brownian sheet. The new SPDE allows us to prove the strong Markov property of the pair $(u,\\partial_x u)$ when seen as a pro"},"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":"1305.3325","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-05-14T23:51:23Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"d501f8054b4365c2001a60627e231d0873501f14e71eb8c457587b67b7b93b6f","abstract_canon_sha256":"3dc4a064347eea08cbf279225442468c208788b20f8d1f6b0c178f2498588dfe"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:25:36.166969Z","signature_b64":"pO6af7NSm7QTlmdVRlVZXHEVU0bRr7KKlUa0a8pU7Tt4rFSw5yQ7pd5mwgBAQduXb0IuQcTqRLdft4d6AnQ3BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a7a1a9566cc6c83bb64a58a93b1aaa2ff93d885fd1fd8c08eb5eb2f4bd7f74bf","last_reissued_at":"2026-05-18T03:25:36.166392Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:25:36.166392Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the spatial dynamics of the solution to the stochastic heat equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.PR","authors_text":"James Bichard, Sigurd Assing","submitted_at":"2013-05-14T23:51:23Z","abstract_excerpt":"We consider the solution of $\\partial_t u=\\partial_x^2 u+\\partial_x\\partial_t B,\\,(x,t)\\in R\\times(0,\\infty)$, subject to $u(x,0)=0,\\,x\\in R$, where $B$ is a Brownian sheet. We show that $u$ also satisfies $\\partial_x^2 u +[\\,(-\\partial_t^2)^{1/2}+\\sqrt{2}\\partial_x(-\\partial_t^2)^{1/4}\\,]\\,u^a= \\partial_x\\partial_t{\\tilde B}$ in $R\\times(0,\\infty)$ where $u^a$ stands for the extension of $u(x,t)$ to $(x,t)\\in R^2$ which is antisymmetric in $t$ and $\\tilde{B}$ is another Brownian sheet. 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