{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:XX42YRXWJEBF23EQORBMAXOY7R","short_pith_number":"pith:XX42YRXW","canonical_record":{"source":{"id":"1412.8732","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-12-30T19:02:23Z","cross_cats_sorted":[],"title_canon_sha256":"88bdbf6f00b519378bf11b3479a161063a95dcb896a5e314112f8d43d0794a14","abstract_canon_sha256":"83b875f00cdbf842577031e23c84412ddc3c5490f7c8ce6ec3e5de1ba4ee4d0b"},"schema_version":"1.0"},"canonical_sha256":"bdf9ac46f649025d6c907442c05dd8fc6759a516e6723f2117fdfc45e60ae26e","source":{"kind":"arxiv","id":"1412.8732","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1412.8732","created_at":"2026-05-18T00:29:39Z"},{"alias_kind":"arxiv_version","alias_value":"1412.8732v3","created_at":"2026-05-18T00:29:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1412.8732","created_at":"2026-05-18T00:29:39Z"},{"alias_kind":"pith_short_12","alias_value":"XX42YRXWJEBF","created_at":"2026-05-18T12:28:57Z"},{"alias_kind":"pith_short_16","alias_value":"XX42YRXWJEBF23EQ","created_at":"2026-05-18T12:28:57Z"},{"alias_kind":"pith_short_8","alias_value":"XX42YRXW","created_at":"2026-05-18T12:28:57Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:XX42YRXWJEBF23EQORBMAXOY7R","target":"record","payload":{"canonical_record":{"source":{"id":"1412.8732","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-12-30T19:02:23Z","cross_cats_sorted":[],"title_canon_sha256":"88bdbf6f00b519378bf11b3479a161063a95dcb896a5e314112f8d43d0794a14","abstract_canon_sha256":"83b875f00cdbf842577031e23c84412ddc3c5490f7c8ce6ec3e5de1ba4ee4d0b"},"schema_version":"1.0"},"canonical_sha256":"bdf9ac46f649025d6c907442c05dd8fc6759a516e6723f2117fdfc45e60ae26e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:29:39.871263Z","signature_b64":"Nv+bKrqDLlhXODrlGAZ6Q0e8OXj/j5Lg3wMggXLNfLzJpuwV0V0htdYNFR6LDAO9Xevc8hF8OGtzlTSTTZF7AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bdf9ac46f649025d6c907442c05dd8fc6759a516e6723f2117fdfc45e60ae26e","last_reissued_at":"2026-05-18T00:29:39.870554Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:29:39.870554Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1412.8732","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:29:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"r3ZEtuAYMp/iopmHsbMSXUrhBFyM2mNffhEzKTN1WhNcM1jgRRSaeiIVVCo4Ji3yRUAJiVaZyInl0/fKjMFZDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T16:49:29.352545Z"},"content_sha256":"6f2ac4bf9a829204cedc66e9ef733817ef22ac890c8cb96231d0a97c975201af","schema_version":"1.0","event_id":"sha256:6f2ac4bf9a829204cedc66e9ef733817ef22ac890c8cb96231d0a97c975201af"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:XX42YRXWJEBF23EQORBMAXOY7R","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Parametrix construction of the transition probability density of the solution to an SDE driven by $\\alpha$-stable noise","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Alexei Kulik, Victoria Knopova","submitted_at":"2014-12-30T19:02:23Z","abstract_excerpt":"Let $L:= -a(x) (-\\Delta)^{\\alpha/2}+ (b(x), \\nabla)$, where $\\alpha\\in (0,2)$, and $a:\\rd\\to (0,\\infty)$, $b: \\rd\\to \\rd$. Under certain regularity assumptions on the coefficients $a$ and $b$, we associate with the $C_\\infty(\\rd)$-closure of $(L, C_\\infty^2(\\rd))$ a Feller Markov process $X$, which possesses a transition probability density $p_t(x,y)$.\n  To construct this transition probability density and to obtain the two-sided estimates on it, we develop a new version of the parametrix method, which allows us to handle the case $0<\\alpha\\leq 1$ and $b\\neq 0$, i.e. when the gradient part of "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.8732","kind":"arxiv","version":3},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:29:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3OaZR0AWyRjLQltMa0E3bhxo14/+3W6ijqvRulPsS2pndg4haKVXEdl86jl4icMLKRx9MG1dwWPSE8vqPqLJAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T16:49:29.352901Z"},"content_sha256":"b1e040adb71dd5852495fe9646429c06909543c94f62ff1177fdcba3187d70b6","schema_version":"1.0","event_id":"sha256:b1e040adb71dd5852495fe9646429c06909543c94f62ff1177fdcba3187d70b6"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XX42YRXWJEBF23EQORBMAXOY7R/bundle.json","state_url":"https://pith.science/pith/XX42YRXWJEBF23EQORBMAXOY7R/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XX42YRXWJEBF23EQORBMAXOY7R/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-26T16:49:29Z","links":{"resolver":"https://pith.science/pith/XX42YRXWJEBF23EQORBMAXOY7R","bundle":"https://pith.science/pith/XX42YRXWJEBF23EQORBMAXOY7R/bundle.json","state":"https://pith.science/pith/XX42YRXWJEBF23EQORBMAXOY7R/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XX42YRXWJEBF23EQORBMAXOY7R/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:XX42YRXWJEBF23EQORBMAXOY7R","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":"83b875f00cdbf842577031e23c84412ddc3c5490f7c8ce6ec3e5de1ba4ee4d0b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-12-30T19:02:23Z","title_canon_sha256":"88bdbf6f00b519378bf11b3479a161063a95dcb896a5e314112f8d43d0794a14"},"schema_version":"1.0","source":{"id":"1412.8732","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1412.8732","created_at":"2026-05-18T00:29:39Z"},{"alias_kind":"arxiv_version","alias_value":"1412.8732v3","created_at":"2026-05-18T00:29:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1412.8732","created_at":"2026-05-18T00:29:39Z"},{"alias_kind":"pith_short_12","alias_value":"XX42YRXWJEBF","created_at":"2026-05-18T12:28:57Z"},{"alias_kind":"pith_short_16","alias_value":"XX42YRXWJEBF23EQ","created_at":"2026-05-18T12:28:57Z"},{"alias_kind":"pith_short_8","alias_value":"XX42YRXW","created_at":"2026-05-18T12:28:57Z"}],"graph_snapshots":[{"event_id":"sha256:b1e040adb71dd5852495fe9646429c06909543c94f62ff1177fdcba3187d70b6","target":"graph","created_at":"2026-05-18T00:29:39Z","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":"Let $L:= -a(x) (-\\Delta)^{\\alpha/2}+ (b(x), \\nabla)$, where $\\alpha\\in (0,2)$, and $a:\\rd\\to (0,\\infty)$, $b: \\rd\\to \\rd$. Under certain regularity assumptions on the coefficients $a$ and $b$, we associate with the $C_\\infty(\\rd)$-closure of $(L, C_\\infty^2(\\rd))$ a Feller Markov process $X$, which possesses a transition probability density $p_t(x,y)$.\n  To construct this transition probability density and to obtain the two-sided estimates on it, we develop a new version of the parametrix method, which allows us to handle the case $0<\\alpha\\leq 1$ and $b\\neq 0$, i.e. when the gradient part of ","authors_text":"Alexei Kulik, Victoria Knopova","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-12-30T19:02:23Z","title":"Parametrix construction of the transition probability density of the solution to an SDE driven by $\\alpha$-stable noise"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.8732","kind":"arxiv","version":3},"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":"sha256:6f2ac4bf9a829204cedc66e9ef733817ef22ac890c8cb96231d0a97c975201af","target":"record","created_at":"2026-05-18T00:29:39Z","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":"83b875f00cdbf842577031e23c84412ddc3c5490f7c8ce6ec3e5de1ba4ee4d0b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-12-30T19:02:23Z","title_canon_sha256":"88bdbf6f00b519378bf11b3479a161063a95dcb896a5e314112f8d43d0794a14"},"schema_version":"1.0","source":{"id":"1412.8732","kind":"arxiv","version":3}},"canonical_sha256":"bdf9ac46f649025d6c907442c05dd8fc6759a516e6723f2117fdfc45e60ae26e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bdf9ac46f649025d6c907442c05dd8fc6759a516e6723f2117fdfc45e60ae26e","first_computed_at":"2026-05-18T00:29:39.870554Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:29:39.870554Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Nv+bKrqDLlhXODrlGAZ6Q0e8OXj/j5Lg3wMggXLNfLzJpuwV0V0htdYNFR6LDAO9Xevc8hF8OGtzlTSTTZF7AQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:29:39.871263Z","signed_message":"canonical_sha256_bytes"},"source_id":"1412.8732","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6f2ac4bf9a829204cedc66e9ef733817ef22ac890c8cb96231d0a97c975201af","sha256:b1e040adb71dd5852495fe9646429c06909543c94f62ff1177fdcba3187d70b6"],"state_sha256":"244b90f22911f2ce4c0633521d43f1c5206b2d2d0cc4bd0787f2b8c1aa707a0d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"MPTKYP4CbTRsfdBYlFJt07gIqNv7aM/QDmRbndSXSxT2dcmwvicX+xBvPkWtpm+Ury4OHispnLqm0B7DbYxxBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-26T16:49:29.354881Z","bundle_sha256":"367bc9cddd22aa0fd9bcfe365e705cac571c85481f31d783c8223926d1674965"}}