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We also prove that in the contrary case of walk dimension two, i.e., where full off-diagonal \\emph{Gaussian} estimates of the heat kernel hold, the symmetric measure and the energy m"},"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":"1910.02601","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2019-10-07T04:18:58Z","cross_cats_sorted":[],"title_canon_sha256":"c3adbb86f8419fe79786dc6322e97bc561ced5db288b8e7e09ef00feec338326","abstract_canon_sha256":"06e8f0b64623834a7dcc4af84487aeb26d9471110254be9cabdad29131c96583"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T01:34:20.495823Z","signature_b64":"pa500Oc3sZygHZ9wQzbUoamjEQsgaUjBdTVpWQ+o5R7GDATh1zMxVhjhcy0JfJGNH5YoL6bzx/dGscRsu67/AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"873ff6358dba7b7654d0398000decd50b2ae914b9499b07ea93912c9dbaeb4ca","last_reissued_at":"2026-07-05T01:34:20.495472Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T01:34:20.495472Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On singularity of energy measures for symmetric diffusions with full off-diagonal heat kernel estimates","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Mathav Murugan, Naotaka Kajino","submitted_at":"2019-10-07T04:18:58Z","abstract_excerpt":"We show that for a strongly local, regular symmetric Dirichlet form over a complete, locally compact geodesic metric space, full off-diagonal heat kernel estimates with walk dimension strictly larger than two (\\emph{sub-Gaussian} estimates) imply the singularity of the energy measures with respect to the symmetric measure, verifying a conjecture by M.\\ T.\\ Barlow in [Contemp.\\ Math., vol.\\ 338, 2003, pp.\\ 11--40]. 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