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The solution of the Cauchy problem above coincides with the distribution of the n-dimensional process \\bm{S}_n^{2\\beta} \\mathcal{L} c^2 {L}^{\\nu_1,..., \\nu_m} (t) \\r, t>0, where \\bm{S}_n^{2\\beta} is an isotropic stable process independent from {L}^{\\nu_1,..., \\nu_m}(t) which is the inverse of {H}^{\\nu"},"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":"1207.3284","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-07-13T15:44:40Z","cross_cats_sorted":[],"title_canon_sha256":"cd0465db234bca0f7e2e9bf9f741027a572a6ccb93466601aefe7e2c1900862f","abstract_canon_sha256":"97f12a78a0a59c225d72cff77984da3bdba21d2b65db705fb0bab76c5bfba639"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:33:24.711471Z","signature_b64":"0HQmplyCgNy6C5/TV4cjcOysVvNNU6vO+O2ieZ3q+YmnVfQHJLePDDkh04s58BpzsnJjPzgJsH/p9ec9BzMdCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"db881e9db3b82019ca77c2bd1ad3a38b0dce4612cc5f012926323ed41302bdfb","last_reissued_at":"2026-05-18T01:33:24.710613Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:33:24.710613Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Space-time fractional equations and the related stable processes at random time","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Bruno Toaldo, Enzo Orsingher","submitted_at":"2012-07-13T15:44:40Z","abstract_excerpt":"In this paper we consider the general fractional equation \\sum_{j=1}^m \\lambda_j \\frac{\\partial^{\\nu_j}}{\\partial t^{\\nu_j}} w(x_1,..., x_n ; t) = -c^2 (-\\Delta)^\\beta w(x_1,..., x_n ; t), for \\nu_j \\in (0,1], \\beta \\in (0,1] with initial condition w(x_1,..., x_n ; 0)= \\prod_{j=1}^n \\delta (x_j). The solution of the Cauchy problem above coincides with the distribution of the n-dimensional process \\bm{S}_n^{2\\beta} \\mathcal{L} c^2 {L}^{\\nu_1,..., \\nu_m} (t) \\r, t>0, where \\bm{S}_n^{2\\beta} is an isotropic stable process independent from {L}^{\\nu_1,..., \\nu_m}(t) which is the inverse of {H}^{\\nu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.3284","kind":"arxiv","version":2},"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":"1207.3284","created_at":"2026-05-18T01:33:24.710761+00:00"},{"alias_kind":"arxiv_version","alias_value":"1207.3284v2","created_at":"2026-05-18T01:33:24.710761+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.3284","created_at":"2026-05-18T01:33:24.710761+00:00"},{"alias_kind":"pith_short_12","alias_value":"3OEB5HNTXAQB","created_at":"2026-05-18T12:26:53.410803+00:00"},{"alias_kind":"pith_short_16","alias_value":"3OEB5HNTXAQBTSTX","created_at":"2026-05-18T12:26:53.410803+00:00"},{"alias_kind":"pith_short_8","alias_value":"3OEB5HNT","created_at":"2026-05-18T12:26:53.410803+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/3OEB5HNTXAQBTSTXYK6RVU5DRM","json":"https://pith.science/pith/3OEB5HNTXAQBTSTXYK6RVU5DRM.json","graph_json":"https://pith.science/api/pith-number/3OEB5HNTXAQBTSTXYK6RVU5DRM/graph.json","events_json":"https://pith.science/api/pith-number/3OEB5HNTXAQBTSTXYK6RVU5DRM/events.json","paper":"https://pith.science/paper/3OEB5HNT"},"agent_actions":{"view_html":"https://pith.science/pith/3OEB5HNTXAQBTSTXYK6RVU5DRM","download_json":"https://pith.science/pith/3OEB5HNTXAQBTSTXYK6RVU5DRM.json","view_paper":"https://pith.science/paper/3OEB5HNT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1207.3284&json=true","fetch_graph":"https://pith.science/api/pith-number/3OEB5HNTXAQBTSTXYK6RVU5DRM/graph.json","fetch_events":"https://pith.science/api/pith-number/3OEB5HNTXAQBTSTXYK6RVU5DRM/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/3OEB5HNTXAQBTSTXYK6RVU5DRM/action/timestamp_anchor","attest_storage":"https://pith.science/pith/3OEB5HNTXAQBTSTXYK6RVU5DRM/action/storage_attestation","attest_author":"https://pith.science/pith/3OEB5HNTXAQBTSTXYK6RVU5DRM/action/author_attestation","sign_citation":"https://pith.science/pith/3OEB5HNTXAQBTSTXYK6RVU5DRM/action/citation_signature","submit_replication":"https://pith.science/pith/3OEB5HNTXAQBTSTXYK6RVU5DRM/action/replication_record"}},"created_at":"2026-05-18T01:33:24.710761+00:00","updated_at":"2026-05-18T01:33:24.710761+00:00"}