{"paper":{"title":"Systems of stochastic Poisson equations: hitting probabilities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Marta Sanz-Sol\\'e, No\\`elia Viles","submitted_at":"2016-12-14T10:42:31Z","abstract_excerpt":"We consider a $d$-dimensional random field $u=(u(x), x\\in D)$ that solves a system of elliptic stochastic equations on a bounded domain $D\\subset \\mathbb{R}^k$, with additive white noise and spatial dimension $k=1,2,3$. Properties of $u$ and its probability law are proved. For Gaussian solutions, using results from [Dalang and Sanz-Sol\\'e, 2009], we establish upper and lower bounds on hitting probabilities in terms of the Hausdorff measure and Bessel-Riesz capacity, respectively. This relies on precise estimates on the canonical distance of the process or, equivalently, on $L^2$ estimates of i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.04567","kind":"arxiv","version":3},"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"}