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It is proved that for any $t>0$, conditionally on $\\omega(\\cdot)$,\n  \\label{*} \\mathbb{E}_0 \\exp\\{\\theta \\int_0^t \\bar{V}(B_s) ds\\} \\  < \\infty \\ a.s. & \\text{if} \\theta< 1/16, \\medskip\n  = \\infty \\ a.s. & \\text{if} \\theta> 1/16, where $\\bar{V}(x)$ is the renormalized Poisson potential\n  $$ \\bar{V}(x)=\\int_{\\mathbb{R}^3} \\frac{1}{| x-y |^2} \\big[\\omega(dy)-dy\\big]. $$ Then the long term behavior of the quenched exponential moment \\eqref{*} is determined for $\\theta \\in (0, 1/16)"},"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":"1103.5717","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-03-29T17:37:07Z","cross_cats_sorted":[],"title_canon_sha256":"6cdb614e780a5c8dab2a942063e3e895427abb9bc63460b7e024cc6238a81997","abstract_canon_sha256":"029f9acd884dfa222b8b426fbdde369d5b5d38e39933837affae6657210ac8de"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:25:29.022986Z","signature_b64":"1IfBt/WqkLM/irCFmXQALJai4PFnLG3pH18v9cHUJ/hbtEPKtHQ0V+6mXItg9Wqt/oTGTCRSXYmAXBQNgHRDCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8dc87ba13865e51eb3c1ad3616a967caae79280be75c5557d81c71e0eb882e0d","last_reissued_at":"2026-05-18T04:25:29.022537Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:25:29.022537Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Spatial Brownian motion in renormalized Poisson potential: A critical case","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Jan Rosinski, Xia Chen","submitted_at":"2011-03-29T17:37:07Z","abstract_excerpt":"Let $B_s$ be a three dimensional Brownian motion and $\\omega(dx)$ be an independent Poisson field on $\\mathbb{R}^3$. It is proved that for any $t>0$, conditionally on $\\omega(\\cdot)$,\n  \\label{*} \\mathbb{E}_0 \\exp\\{\\theta \\int_0^t \\bar{V}(B_s) ds\\} \\  < \\infty \\ a.s. & \\text{if} \\theta< 1/16, \\medskip\n  = \\infty \\ a.s. & \\text{if} \\theta> 1/16, where $\\bar{V}(x)$ is the renormalized Poisson potential\n  $$ \\bar{V}(x)=\\int_{\\mathbb{R}^3} \\frac{1}{| x-y |^2} \\big[\\omega(dy)-dy\\big]. $$ Then the long term behavior of the quenched exponential moment \\eqref{*} is determined for $\\theta \\in (0, 1/16)"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.5717","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1103.5717","created_at":"2026-05-18T04:25:29.022599+00:00"},{"alias_kind":"arxiv_version","alias_value":"1103.5717v1","created_at":"2026-05-18T04:25:29.022599+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1103.5717","created_at":"2026-05-18T04:25:29.022599+00:00"},{"alias_kind":"pith_short_12","alias_value":"RXEHXIJYMXSR","created_at":"2026-05-18T12:26:41.206345+00:00"},{"alias_kind":"pith_short_16","alias_value":"RXEHXIJYMXSR5M6B","created_at":"2026-05-18T12:26:41.206345+00:00"},{"alias_kind":"pith_short_8","alias_value":"RXEHXIJY","created_at":"2026-05-18T12:26:41.206345+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/RXEHXIJYMXSR5M6BVU3BNKLHZK","json":"https://pith.science/pith/RXEHXIJYMXSR5M6BVU3BNKLHZK.json","graph_json":"https://pith.science/api/pith-number/RXEHXIJYMXSR5M6BVU3BNKLHZK/graph.json","events_json":"https://pith.science/api/pith-number/RXEHXIJYMXSR5M6BVU3BNKLHZK/events.json","paper":"https://pith.science/paper/RXEHXIJY"},"agent_actions":{"view_html":"https://pith.science/pith/RXEHXIJYMXSR5M6BVU3BNKLHZK","download_json":"https://pith.science/pith/RXEHXIJYMXSR5M6BVU3BNKLHZK.json","view_paper":"https://pith.science/paper/RXEHXIJY","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1103.5717&json=true","fetch_graph":"https://pith.science/api/pith-number/RXEHXIJYMXSR5M6BVU3BNKLHZK/graph.json","fetch_events":"https://pith.science/api/pith-number/RXEHXIJYMXSR5M6BVU3BNKLHZK/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/RXEHXIJYMXSR5M6BVU3BNKLHZK/action/timestamp_anchor","attest_storage":"https://pith.science/pith/RXEHXIJYMXSR5M6BVU3BNKLHZK/action/storage_attestation","attest_author":"https://pith.science/pith/RXEHXIJYMXSR5M6BVU3BNKLHZK/action/author_attestation","sign_citation":"https://pith.science/pith/RXEHXIJYMXSR5M6BVU3BNKLHZK/action/citation_signature","submit_replication":"https://pith.science/pith/RXEHXIJYMXSR5M6BVU3BNKLHZK/action/replication_record"}},"created_at":"2026-05-18T04:25:29.022599+00:00","updated_at":"2026-05-18T04:25:29.022599+00:00"}