{"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. 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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.09972","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}