{"paper":{"title":"Smoothness of the density for McKean-Vlasov SDEs with measurable kernel","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Yi Han","submitted_at":"2022-08-04T16:59:39Z","abstract_excerpt":"Consider the McKean-Vlasov SDE $$\n  dX_t=\\langle b(X_t-\\cdot),\\mu_t\\rangle dt+dW_t,\\quad \\mu_t=\\operatorname{Law}(X_t), $$ where $W$ is the $n$-dimensional Brownian motion and $b:\\mathbb{R}^d\\to\\mathbb{R}^d$ is a measurable function. First assuming $b\\in L^\\infty$, we prove that the law $\\mu_t$ of $X_t$ has a density $p_t$ with respect to the Lebesgue measure, which is continuously differentiable with gradient being $\\gamma$-H\\\"older continuous for each $\\gamma\\in(0,1)$. Assume further that $b\\in \\mathcal{C}_b^1$, we prove that the density $p_t$ is infinitely differentiable. In the regularizat"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2208.02771","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2208.02771/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}