{"paper":{"title":"On some smoothening effects of the transition semigroup of a L\\'evy process","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Lihu Xu, Szymon Peszat, Zhao Dong","submitted_at":"2014-07-28T23:58:20Z","abstract_excerpt":"Let $(P_t)$ be the transition semigroup of a L\\'evy process $L$ taking values in a Hilbert space $H$. Let $\\nu$ be the L\\'evy measure of $L$. It is shown that for any bounded and measurable function $f$, $$ \\int_H\\left\\vert P_tf(x+y)-P_tf(x)\\right\\vert ^2 \\nu (\\dif y)\\le \\frac 1 t P_tf^2(x) \\qquad \\text{for all $t>0$, $x\\in H$.} $$ As $\\nu$ can be infinite this formula establishes some smoothening effect of the semigroup $(P_t)$. In the paper some applications of the formula will be presented as well."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.7603","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"}