{"paper":{"title":"Optimal Strong Rates of Convergence for a Space-Time Discretization of the Stochastic Allen-Cahn Equation with multiplicative noise","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Ananta K. Majee, Andreas Prohl","submitted_at":"2017-05-28T22:19:44Z","abstract_excerpt":"The stochastic Allen-Cahn equation with multiplicative noise involves the nonlinear drift operator ${\\mathscr A}(x) = \\Delta x - \\bigl(\\vert x\\vert^2 -1\\bigr)x$. We use the fact that ${\\mathscr A}(x) = -{\\mathcal J}^{\\prime}(x)$ satisfies a weak monotonicity property to deduce uniform bounds in strong norms for solutions of the temporal, as well as of the spatio-temporal discretization of the problem. This weak monotonicity property then allows for the estimate $ \\underset{1 \\leq j \\leq J}\\sup {\\mathbb E}\\bigl[ \\Vert X_{t_j} - Y^j\\Vert_{{\\mathbb L}^2}^2\\bigr] \\leq C_{\\delta}(k^{1-\\delta} + h^2"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.09997","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"}