{"paper":{"title":"Resonant averaging for weakly nonlinear stochastic Schr\\\"odinger equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Alberto Maiocchi, Sergei Kuksin","submitted_at":"2013-09-19T15:26:39Z","abstract_excerpt":"We consider the free linear Schroedinger equation on a torus $\\mathbb T^d$, perturbed by a Hamiltonian nonlinearity, driven by a random force and damped by a linear damping: $$u_t -i\\Delta u +i\\nu \\rho |u|^{2q_*}u = - \\nu f(-\\Delta) u + \\sqrt\\nu\\,\\frac{d}{d t}\\sum_{k\\in \\mathbb Z^d} b_k\\beta^k(t)e^{ik\\cdot x} \\ . $$ Here $u=u(t,x),\\ x\\in\\mathbb T^d$, $0<\\nu\\ll1$, $q_*\\in\\mathbb N\\cup\\{0\\}$, $f$ is a positive continuous function, $\\rho$ is a positive parameter and $\\beta^k(t)$ are standard independent complex Wiener processes. We are interested in limiting, as $\\nu\\to0$, behaviour of solutions "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.5022","kind":"arxiv","version":5},"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"}