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These operators are degenerate elliptic of the form $\\mathcal{L}=\\mathcal{L}^{\\sigma,b}+\\mathcal{L}^\\mu$ for some local part $\\mathcal{L}^{\\sigma,b}[u]=\\text{tr}(\\sigma \\sigma^{\\texttt{T}} D^2u)+b \\cdot Du$ and nonlocal part $$ \\mathcal{L}^\\mu[u](x)=\\int \\big(u(x+z)-u(x)-z \\cdot Du(x) \\mathbf{1}_{|z| \\leq 1}\\big) \\, \\text{d} \\mu(z), $$ where $\\mu \\geq 0$ is a so-called L\\'evy measure possibly unbounded for small $z$"},"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":"1807.01843","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-07-05T04:31:27Z","cross_cats_sorted":[],"title_canon_sha256":"68ea417c651c5ce0c7d44f77426a4f7b34d63ec5ae0ae90168da188ba6dda3b6","abstract_canon_sha256":"8b6735800bceb7ef7f849669943aef7f87e6662016e828efe86c8894d929aca0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:11:28.508853Z","signature_b64":"7DRGlRyeKQeAzUpwGXKcV4iaG4DNBqmE0/6ufb4VaEUiwHd1uuIKUALE57nGo2GSN8GmRoUFg9cJ0KmL3ndpCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ffebf42db53fffeb258f97d0d0fc2bee0beb308968d5e7dd6adeb60945952637","last_reissued_at":"2026-05-18T00:11:28.508515Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:11:28.508515Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Characterization of nonlocal diffusion operators satisfying the Liouville theorem. Irrational numbers and subgroups of $\\mathbb{R}^d$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Espen R. Jakobsen, F\\'elix del Teso, J{\\o}rgen Endal, Natha\\\"el Alibaud","submitted_at":"2018-07-05T04:31:27Z","abstract_excerpt":"We investigate the characterization of generators $\\mathcal{L}$ of L\\'evy processes satisfying the Liouville theorem: Bounded functions $u$ solving $\\mathcal{L}[u]=0$ are constant. These operators are degenerate elliptic of the form $\\mathcal{L}=\\mathcal{L}^{\\sigma,b}+\\mathcal{L}^\\mu$ for some local part $\\mathcal{L}^{\\sigma,b}[u]=\\text{tr}(\\sigma \\sigma^{\\texttt{T}} D^2u)+b \\cdot Du$ and nonlocal part $$ \\mathcal{L}^\\mu[u](x)=\\int \\big(u(x+z)-u(x)-z \\cdot Du(x) \\mathbf{1}_{|z| \\leq 1}\\big) \\, \\text{d} \\mu(z), $$ where $\\mu \\geq 0$ is a so-called L\\'evy measure possibly unbounded for small $z$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.01843","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":"1807.01843","created_at":"2026-05-18T00:11:28.508569+00:00"},{"alias_kind":"arxiv_version","alias_value":"1807.01843v1","created_at":"2026-05-18T00:11:28.508569+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1807.01843","created_at":"2026-05-18T00:11:28.508569+00:00"},{"alias_kind":"pith_short_12","alias_value":"77V7ILNVH776","created_at":"2026-05-18T12:32:11.075285+00:00"},{"alias_kind":"pith_short_16","alias_value":"77V7ILNVH776WJMP","created_at":"2026-05-18T12:32:11.075285+00:00"},{"alias_kind":"pith_short_8","alias_value":"77V7ILNV","created_at":"2026-05-18T12:32:11.075285+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/77V7ILNVH776WJMPS7INB7BL5Y","json":"https://pith.science/pith/77V7ILNVH776WJMPS7INB7BL5Y.json","graph_json":"https://pith.science/api/pith-number/77V7ILNVH776WJMPS7INB7BL5Y/graph.json","events_json":"https://pith.science/api/pith-number/77V7ILNVH776WJMPS7INB7BL5Y/events.json","paper":"https://pith.science/paper/77V7ILNV"},"agent_actions":{"view_html":"https://pith.science/pith/77V7ILNVH776WJMPS7INB7BL5Y","download_json":"https://pith.science/pith/77V7ILNVH776WJMPS7INB7BL5Y.json","view_paper":"https://pith.science/paper/77V7ILNV","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1807.01843&json=true","fetch_graph":"https://pith.science/api/pith-number/77V7ILNVH776WJMPS7INB7BL5Y/graph.json","fetch_events":"https://pith.science/api/pith-number/77V7ILNVH776WJMPS7INB7BL5Y/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/77V7ILNVH776WJMPS7INB7BL5Y/action/timestamp_anchor","attest_storage":"https://pith.science/pith/77V7ILNVH776WJMPS7INB7BL5Y/action/storage_attestation","attest_author":"https://pith.science/pith/77V7ILNVH776WJMPS7INB7BL5Y/action/author_attestation","sign_citation":"https://pith.science/pith/77V7ILNVH776WJMPS7INB7BL5Y/action/citation_signature","submit_replication":"https://pith.science/pith/77V7ILNVH776WJMPS7INB7BL5Y/action/replication_record"}},"created_at":"2026-05-18T00:11:28.508569+00:00","updated_at":"2026-05-18T00:11:28.508569+00:00"}