{"paper":{"title":"Densities for Ornstein-Uhlenbeck processes with jumps","license":"","headline":"","cross_cats":["math.AP"],"primary_cat":"math.PR","authors_text":"Enrico Priola, Jerzy Zabczyk","submitted_at":"2007-08-08T11:38:18Z","abstract_excerpt":"We consider an Ornstein-Uhlenbeck process with values in R^n driven by a L\\'evy process (Z_t) taking values in R^d with d possibly smaller than n. The L\\'evy noise can have a degenerate or even vanishing Gaussian component.\n  Under a controllability condition and an assumption on the L\\'evy measure of (Z_t), we prove that the law of the Ornstein-Uhlenbeck process at any time t>0 has a density on R^n. Moreover, when the L\\'evy process is of $\\alpha$-stable type, $\\alpha \\in (0,2)$, we show that such density is a $C^{\\infty}$-function."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0708.1084","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"}