{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2006:B62Z43KO47AIO2GDO7UYPU5IX6","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":"177e551689f537f622738a996d1a23625fddcf2335f35e66b974dc42454c6fa5","cross_cats_sorted":[],"license":"","primary_cat":"math.PR","submitted_at":"2006-09-25T07:00:49Z","title_canon_sha256":"fcbaab00695ab7927f15d0a2f26315643b2f7f34b1908e9d5d24580f1dba1f63"},"schema_version":"1.0","source":{"id":"math/0609679","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0609679","created_at":"2026-05-18T01:08:49Z"},{"alias_kind":"arxiv_version","alias_value":"math/0609679v1","created_at":"2026-05-18T01:08:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0609679","created_at":"2026-05-18T01:08:49Z"},{"alias_kind":"pith_short_12","alias_value":"B62Z43KO47AI","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_16","alias_value":"B62Z43KO47AIO2GD","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_8","alias_value":"B62Z43KO","created_at":"2026-05-18T12:25:53Z"}],"graph_snapshots":[{"event_id":"sha256:abf0a7fecd1ad82a584e7ad2ab89b4f0013646844744ac60ea37bf4f5238e842","target":"graph","created_at":"2026-05-18T01:08:49Z","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":"Dunkl processes are martingales as well as c\\`{a}dl\\`{a}g homogeneous Markov processes taking values in $\\mathbb{R}^d$ and they are naturally associated with a root system. In this paper we study the jumps of these processes, we describe precisely their martingale decompositions into continuous and purely discontinuous parts and we obtain a Wiener chaos decomposition of the corresponding $L^2$ spaces of these processes in terms of adequate mixed multiple stochastic integrals.","authors_text":"L\\'eonard Gallardo, Marc Yor","cross_cats":[],"headline":"","license":"","primary_cat":"math.PR","submitted_at":"2006-09-25T07:00:49Z","title":"A chaotic representation property of the multidimensional Dunkl processes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0609679","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":"sha256:9a1eb6cd6157a823d9fc71617403dc9b05b4e0a27f4e83dab739b2a3d9ba2c9b","target":"record","created_at":"2026-05-18T01:08:49Z","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":"177e551689f537f622738a996d1a23625fddcf2335f35e66b974dc42454c6fa5","cross_cats_sorted":[],"license":"","primary_cat":"math.PR","submitted_at":"2006-09-25T07:00:49Z","title_canon_sha256":"fcbaab00695ab7927f15d0a2f26315643b2f7f34b1908e9d5d24580f1dba1f63"},"schema_version":"1.0","source":{"id":"math/0609679","kind":"arxiv","version":1}},"canonical_sha256":"0fb59e6d4ee7c08768c377e987d3a8bfaacf592ec080b1fced3eb8229a430160","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0fb59e6d4ee7c08768c377e987d3a8bfaacf592ec080b1fced3eb8229a430160","first_computed_at":"2026-05-18T01:08:49.695099Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:08:49.695099Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"/a7Hi/G5UNfrfpzBUSkM38psVfXX8P3nM5HMoc4NSxastJ3VB/Y5W+NZLHE9yecImE9qStH4TXzlPS3WJOuRCw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:08:49.695726Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0609679","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9a1eb6cd6157a823d9fc71617403dc9b05b4e0a27f4e83dab739b2a3d9ba2c9b","sha256:abf0a7fecd1ad82a584e7ad2ab89b4f0013646844744ac60ea37bf4f5238e842"],"state_sha256":"0557bc5cbb4c150159dbf8411ff830771df98fae92fc192b9a2a6c684857b027"}