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Under a uniform H\\\"older continuous assumptions on $\\alpha(x)$ and $x\\mapsto \\kappa(x,z)$, we obtain existence, upper and lower bounds, and regularity properties of the heat kernel associated with the following non-local operator of variable order $$ \\LL f(x)=\\int_{\\R^d}\\big(f(x+z)-f(x)-\\langle\\nabla f(x), z\\rangle \\I_{\\{|z|\\le 1\\}}\\big) \\frac{\\kappa(x,z)}{|z|^{d+\\alpha(x"},"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":"1811.09972","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-11-25T08:27:00Z","cross_cats_sorted":[],"title_canon_sha256":"6ff98fd274fb10b4cd970baaa9e7c2bb2bebe560e029ba1d582a9e3bbb0650cb","abstract_canon_sha256":"df8851133ab9a5aa6379641fb4662abb19a789e9a36ac6ae70561379da279390"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:59:58.970629Z","signature_b64":"zr1OXpw3V7bNweSNHuBoHWImUHewcfnVyj3iyu5bi/nkvk/y/aLRxQuB6qkGCiTZ2S9KURdtkPiddmwKhIuvCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5a9bc0ad7ba749766473dba9bc709767930936f1e4a8935838d8167251b56d4e","last_reissued_at":"2026-05-17T23:59:58.970013Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:59:58.970013Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Heat kernel for non-local operators with variable order","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Jian Wang, Xin Chen, Zhen-Qing Chen","submitted_at":"2018-11-25T08:27:00Z","abstract_excerpt":"Let $\\alpha(x)$ be a measurable function taking values in $ [\\alpha_1,\\alpha_2]$ for $0<\\A_1\\le \\A_2<2$, and $\\kappa(x,z)$ be a positive measurable function that is symmetric in $z$ and bounded between two positive constants. 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