{"paper":{"title":"Exponential loss of memory for the 2-dimensional Allen-Cahn equation with small noise","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Hendrik Weber, Pavlos Tsatsoulis","submitted_at":"2018-08-13T12:34:10Z","abstract_excerpt":"We prove an asymptotic coupling theorem for the $2$-dimensional Allen--Cahn equation perturbed by a small space-time white noise. We show that with overwhelming probability two profiles that start close to the minimisers of the potential of the deterministic system contract exponentially fast in a suitable topology. In the $1$-dimensional case a similar result was shown in \\cite{MS88,MOS89}.\n  It is well-known that in more than one dimension solutions of this equation are distribution-valued, and the equation has to be interpreted in a renormalised sense. Formally, this renormalisation corresp"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.04171","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"}