{"paper":{"title":"Invariant Measure of the Camassa-Holm Equation with Linear Multiplicative Noise","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.AP","authors_text":"Pei Zheng, Wei Luo, Zhaoyang Yin","submitted_at":"2026-07-02T02:29:01Z","abstract_excerpt":"In this paper, we prove that the solution map of Camassa-Holm equation with linear multiplicative noise\n  $$\n  \\left\\{\n  \\begin{array}{l}\n  {\\rm d}u+(u\\partial_xu+\\partial_xP[u])\\,{\\rm d}t=\\beta u\\,{\\rm d}W,\n  u(0,x)=u_0(x),\n  P[u]=(1-\\partial_x^2)^{-1}\\left(u^2+\\frac 1 2(\\partial_x u)^2\\right)\n  \\end{array}\n  \\right.\n  $$\n  depends almost surely continuously on the deterministic initial data in $H^s$ for $s>3/2$. Furthermore, we prove the existence and non-uniqueness of an invariant measure for the Camassa-Holm equation with linear multiplicative noise."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.01611","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.01611/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}