{"paper":{"title":"On the probabilistic Cauchy theory of the cubic nonlinear Schr\\\"odinger equation on $\\mathbb R^d$, $d \\geq 3$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"\\'Arp\\'ad B\\'enyi, Oana Pocovnicu, Tadahiro Oh","submitted_at":"2014-05-28T18:38:15Z","abstract_excerpt":"We consider the Cauchy problem of the cubic nonlinear Schr\\\"odinger equation (NLS) on $\\mathbb R^d$, $d \\geq 3$, with random initial data and prove almost sure well-posedness results below the scaling critical regularity $s_\\text{crit} = \\frac{d-2}{2}$. More precisely, given a function on $\\mathbb R^d$, we introduce a randomization adapted to the Wiener decomposition, and, intrinsically, to the so-called modulation spaces. Our goal in this paper is three-fold. (i) We prove almost sure local well-posedness of the cubic NLS below the scaling critical regularity along with small data global exist"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.7327","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"}