{"paper":{"title":"Stochastic quasi-geostrophic equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Michael R\\\"ockner, Rongchan Zhu, Xiangchan Zhu","submitted_at":"2011-08-24T17:41:39Z","abstract_excerpt":"In this note we study the 2d stochastic quasi-geostrophic equation in $\\mathbb{T}^2$ for general parameter $\\alpha\\in (0,1)$ and multiplicative noise. We prove the existence of martingale solutions and pathwise uniqueness under some condition in the general case, i.e. for all $\\alpha\\in (0,1)$. In the subcritical case $\\alpha>1/2$, we prove existence and uniqueness of (probabilistically) strong solutions and construct a Markov family of solutions. In particular, it is uniquely ergodic for $\\alpha>2/3$ provided the noise is non-degenerate. In this case, the convergence to the (unique) invariant"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.4896","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":""},"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"}