{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:AEZIEVBGQZ732NF55CLNIO4KSQ","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"44cab5abb0c64fa6dce8109d6b7c094de690155b6e024c0210094c13da0da949","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-10-08T15:19:23Z","title_canon_sha256":"173394ea53c8849e2a7efd0652f0a021127705ff5ddf5dd5db6241a45115b156"},"schema_version":"1.0","source":{"id":"1510.02361","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1510.02361","created_at":"2026-05-18T01:30:43Z"},{"alias_kind":"arxiv_version","alias_value":"1510.02361v1","created_at":"2026-05-18T01:30:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.02361","created_at":"2026-05-18T01:30:43Z"},{"alias_kind":"pith_short_12","alias_value":"AEZIEVBGQZ73","created_at":"2026-05-18T12:29:10Z"},{"alias_kind":"pith_short_16","alias_value":"AEZIEVBGQZ732NF5","created_at":"2026-05-18T12:29:10Z"},{"alias_kind":"pith_short_8","alias_value":"AEZIEVBG","created_at":"2026-05-18T12:29:10Z"}],"graph_snapshots":[{"event_id":"sha256:9bffde8c82060be8616e211708736652e99d02eae21f940be2eb5791bfd9474a","target":"graph","created_at":"2026-05-18T01:30:43Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We investigate the large time behavior of solutions to the spatially homogeneous linear Boltzmann equation from a semigroup viewpoint. Our analysis is performed in some (weighted) $L^{1}$-spaces. We deal with both the cases of hard and soft potentials (with angular cut-off). For hard potentials, we provide a new proof of the fact that, in weighted $L^{1}$-spaces with exponential or algebraic weights, the solutions converge exponentially fast towards equilibrium. Our approach uses weak-compactness arguments combined with  recent results of the second author on positive semigroups in $L^{1}$-spa","authors_text":"Bertrand Lods, Mustapha Mokhtar-Kharroubi (LM-Besan\\c{c}on)","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-10-08T15:19:23Z","title":"Convergence to equilibrium for linear spatially homogeneous Boltzmann equation with hard and soft potentials: a semigroup approach in $L^1$-spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.02361","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:af19e01eff7c78bd7bdfd882ac6099eb51feaca50bbf98e617ac93946d09abbb","target":"record","created_at":"2026-05-18T01:30:43Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"44cab5abb0c64fa6dce8109d6b7c094de690155b6e024c0210094c13da0da949","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-10-08T15:19:23Z","title_canon_sha256":"173394ea53c8849e2a7efd0652f0a021127705ff5ddf5dd5db6241a45115b156"},"schema_version":"1.0","source":{"id":"1510.02361","kind":"arxiv","version":1}},"canonical_sha256":"0132825426867fbd34bde896d43b8a9406e78d07fbabdf920a52b8bad26777ad","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0132825426867fbd34bde896d43b8a9406e78d07fbabdf920a52b8bad26777ad","first_computed_at":"2026-05-18T01:30:43.999458Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:30:43.999458Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Mv0DY7rDBc9X/86ZZEg3dtAaf6pzcgFAvXwh2R8GyNp1L1IjNRHcYH/8595ggrx/osKXfcsQ0GJrE567qUBpCw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:30:43.999973Z","signed_message":"canonical_sha256_bytes"},"source_id":"1510.02361","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:af19e01eff7c78bd7bdfd882ac6099eb51feaca50bbf98e617ac93946d09abbb","sha256:9bffde8c82060be8616e211708736652e99d02eae21f940be2eb5791bfd9474a"],"state_sha256":"861c1788110cc036f806bbe6c3247536be3c4f1cd859bd1d08fe0f82bb9d88ae"}