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Let gamma(H,E) denote the space of all gamma-radonifying operators from H to E. We prove that the following assertions are equivalent:\n  (i) the stochastic Cauchy problem dU(t) = AU(t)dt + BdW_H(t) admits an invariant measure on E;\n  (ii) (-A)^{-1/2} B belongs to gamma(H,E);\n  (iii) the Gaussian sum \\sum"},"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":"1206.3656","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-06-16T10:42:53Z","cross_cats_sorted":["math.OC","math.PR"],"title_canon_sha256":"5b650e650696f800eb837bd42b630b42b2394cd3c684fb4764114875baa95f91","abstract_canon_sha256":"2d667b73d666fd9f9bfb7d4f348b02ff624bdde50c25f2b6ab67d5e42bc3ab9b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:57:53.434126Z","signature_b64":"ghFCMG9hJsNwVa/jDNsxkEe9n6Vg3peHCkDyANVJteTl7w1xN6hf92rl0RheRriN97hgggrDN9VtSr+nLg4SDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6357ffcdb2dfec49b2e4582b36cc4f4ad10e565484f935f3bb5a43d1c21c525d","last_reissued_at":"2026-05-18T02:57:53.433362Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:57:53.433362Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The stochastic Weiss conjecture for bounded analytic semigroups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OC","math.PR"],"primary_cat":"math.FA","authors_text":"Bernhard Haak, Jamil Abreu, Jan van Neerven","submitted_at":"2012-06-16T10:42:53Z","abstract_excerpt":"Suppose -A admits a bounded H-infinity calculus of angle less than pi/2 on a Banach space E with Pisier's property (alpha), let B be a bounded linear operator from a Hilbert space H into the extrapolation space E_{-1} of E with respect to A, and let W_H denote an H-cylindrical Brownian motion. Let gamma(H,E) denote the space of all gamma-radonifying operators from H to E. We prove that the following assertions are equivalent:\n  (i) the stochastic Cauchy problem dU(t) = AU(t)dt + BdW_H(t) admits an invariant measure on E;\n  (ii) (-A)^{-1/2} B belongs to gamma(H,E);\n  (iii) the Gaussian sum \\sum"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.3656","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":"1206.3656","created_at":"2026-05-18T02:57:53.433492+00:00"},{"alias_kind":"arxiv_version","alias_value":"1206.3656v1","created_at":"2026-05-18T02:57:53.433492+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.3656","created_at":"2026-05-18T02:57:53.433492+00:00"},{"alias_kind":"pith_short_12","alias_value":"MNL77TNS37WE","created_at":"2026-05-18T12:27:14.488303+00:00"},{"alias_kind":"pith_short_16","alias_value":"MNL77TNS37WETMXE","created_at":"2026-05-18T12:27:14.488303+00:00"},{"alias_kind":"pith_short_8","alias_value":"MNL77TNS","created_at":"2026-05-18T12:27:14.488303+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/MNL77TNS37WETMXELAVTNTCPJL","json":"https://pith.science/pith/MNL77TNS37WETMXELAVTNTCPJL.json","graph_json":"https://pith.science/api/pith-number/MNL77TNS37WETMXELAVTNTCPJL/graph.json","events_json":"https://pith.science/api/pith-number/MNL77TNS37WETMXELAVTNTCPJL/events.json","paper":"https://pith.science/paper/MNL77TNS"},"agent_actions":{"view_html":"https://pith.science/pith/MNL77TNS37WETMXELAVTNTCPJL","download_json":"https://pith.science/pith/MNL77TNS37WETMXELAVTNTCPJL.json","view_paper":"https://pith.science/paper/MNL77TNS","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1206.3656&json=true","fetch_graph":"https://pith.science/api/pith-number/MNL77TNS37WETMXELAVTNTCPJL/graph.json","fetch_events":"https://pith.science/api/pith-number/MNL77TNS37WETMXELAVTNTCPJL/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MNL77TNS37WETMXELAVTNTCPJL/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MNL77TNS37WETMXELAVTNTCPJL/action/storage_attestation","attest_author":"https://pith.science/pith/MNL77TNS37WETMXELAVTNTCPJL/action/author_attestation","sign_citation":"https://pith.science/pith/MNL77TNS37WETMXELAVTNTCPJL/action/citation_signature","submit_replication":"https://pith.science/pith/MNL77TNS37WETMXELAVTNTCPJL/action/replication_record"}},"created_at":"2026-05-18T02:57:53.433492+00:00","updated_at":"2026-05-18T02:57:53.433492+00:00"}