{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:Q7VW76OE2EKYMLQO5O55US2ZSC","short_pith_number":"pith:Q7VW76OE","schema_version":"1.0","canonical_sha256":"87eb6ff9c4d115862e0eebbbda4b5990b56554cd5eb508ac43620cc8d89e8356","source":{"kind":"arxiv","id":"1109.6738","version":4},"attestation_state":"computed","paper":{"title":"Derivative Formula and Gradient Estimates for Gruschin Type Semigroups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Feng-Yu Wang","submitted_at":"2011-09-30T07:51:22Z","abstract_excerpt":"By solving a control problem and using Malliavin calculus, explicit derivative formula is derived for the semigroup $P_t$ generated by the Gruschin type operator on $\\R^{m}\\times \\R^{d}:$ $$L (x,y)=\\ff 1 2 \\bigg\\{\\sum_{i=1}^m \\pp_{x_i}^2 +\\sum_{j,k=1}^d (\\si(x)\\si(x)^*)_{jk} \\pp_{y_j}\\pp_{y_k}\\bigg\\},\\ \\ (x,y)\\in \\R^m\\times\\R^d,$$ where $\\si\\in C^1(\\R^m; \\R^d\\otimes\\R^d)$ might be degenerate. In particular, if $\\si(x)$ is comparable with $|x|^{l}I_{d\\times d}$ for some $l\\ge 1$ in the sense of (\\ref{A4}), then for any $p>1$ there exists a constant $C_p>0$ such that $$|\\nn P_t f(x,y)|\\le \\ff{C_"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1109.6738","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-09-30T07:51:22Z","cross_cats_sorted":[],"title_canon_sha256":"d832789167c250fb74e5f65174469f0203bf7dcfd9becf5aa5a773c0e1c10cd7","abstract_canon_sha256":"fdcf96f7de212c994b0d00cbe2a4ef36070702dd4c519c039350a71d649135f7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:29:14.081561Z","signature_b64":"6C+52YJ7yySnSDLuFTqJcAJpnQ6QY/yfkFqSETIN/Wk/4kjiCmA25CHPNhlT9IA6iqAVLasi+fCxtnwUrxbSDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"87eb6ff9c4d115862e0eebbbda4b5990b56554cd5eb508ac43620cc8d89e8356","last_reissued_at":"2026-05-18T03:29:14.081029Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:29:14.081029Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Derivative Formula and Gradient Estimates for Gruschin Type Semigroups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Feng-Yu Wang","submitted_at":"2011-09-30T07:51:22Z","abstract_excerpt":"By solving a control problem and using Malliavin calculus, explicit derivative formula is derived for the semigroup $P_t$ generated by the Gruschin type operator on $\\R^{m}\\times \\R^{d}:$ $$L (x,y)=\\ff 1 2 \\bigg\\{\\sum_{i=1}^m \\pp_{x_i}^2 +\\sum_{j,k=1}^d (\\si(x)\\si(x)^*)_{jk} \\pp_{y_j}\\pp_{y_k}\\bigg\\},\\ \\ (x,y)\\in \\R^m\\times\\R^d,$$ where $\\si\\in C^1(\\R^m; \\R^d\\otimes\\R^d)$ might be degenerate. 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