{"paper":{"title":"From Gaussian estimates for nonlinear evolution equations to the long time behavior of branching processes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.AP","authors_text":"J. D. Rossi, L. Beznea, L. I. Ignat","submitted_at":"2017-03-08T12:20:07Z","abstract_excerpt":"We study solutions to the evolution equation $u_t=\\Delta u-u +\\sum_{k\\geqslant 1}q_ku^k$, $t>0$, in $\\mathbf{R}^d$. Here the coefficients $q_k\\geqslant 0$ verify $ \\sum_{k\\geqslant 1}q_k=1< \\sum_{k\\geqslant 1}kq_k<\\infty$. First, we deal with existence, uniqueness, and the asymptotic behavior of the solutions as $t\\to +\\infty$. We then deduce results on the long time behavior of the associated branching process, with state space the set of all finite configurations of $\\mathbf{R}^d$, under the assumption that $\\sum_{k\\geq 1} k^2q_k<\\infty$. It turns out that the distribution of the branching p"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.02807","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"}