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Let $r:[0,\\infty[\\to[0,\\infty[$ and $R (t) = \\int_0^t r(s) ds$.\n  We give necessary and sufficient conditions for $\\E_{\\mu} R (\\tau_G)<\\infty$ in terms of the behavior near the origin of the spectral measure of $-A^G.$\n  When $r(t)=t^l$, $l>0$, by means of this condition we derive the Nash inequality for the killed process.\n  In the case of one-dimensional diffusions, this permits to show tha"},"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":"1103.4622","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-03-23T20:30:43Z","cross_cats_sorted":[],"title_canon_sha256":"fe4fc0a7d6afbeda8b6e4c1d9528884837157e3fdcc627d8866d52f98b55afbc","abstract_canon_sha256":"e9800cf65ba9dcb6949433d84685f6065c61b76b73aab5aad994fb0548de5305"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:03:43.393338Z","signature_b64":"2qwQM7l3CmHIkhnE/LWxQVDAK2yqUh/wp0+8Z1PeHfGAo3gYlnRc7+Q+d8J0nS4Qy6dkdng3D/PYKRej3twECQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9a2c0eac588ca01cbcd44f676b837731d54e40eea07cf162ab87004f5b2c490e","last_reissued_at":"2026-05-18T04:03:43.392735Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:03:43.392735Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Spectral condition, hitting times and Nash inequality","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Dasha Loukianova, Eva Loecherbach, Oleg Loukianov","submitted_at":"2011-03-23T20:30:43Z","abstract_excerpt":"Let $X$ be a $\\mu$-symmetric Hunt process on a LCCB space E. For an open set G $\\subseteq$ E, let $\\tau_G$ be the exit time of $X$ from G and $A^G$ be the generator of the process killed when it leaves G. 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