{"paper":{"title":"On properties of a class of strong limits for supercritical superprocesses","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Renming Song, Rui Zhang, Yan-Xia Ren","submitted_at":"2018-03-08T05:26:29Z","abstract_excerpt":"Suppose that $X=\\{X_t, t\\ge 0; \\mathbb{P}_{\\mu}\\}$ is a supercritical superprocess in a locally compact separable metric space $E$. Let $\\phi_0$ be a positive\n  eigenfunction corresponding to the first eigenvalue $\\lambda_0$ of the generator of the mean semigroup of $X$. Then $M_t:=e^{-\\lambda_0t}\\langle\\phi_0, X_t\\rangle$ is a positive martingale. Let $M_\\infty$ be the limit of $M_t$. It is known that $M_\\infty$ is non-degenerate iff the $L\\log L$ condition is satisfied. When the $L\\log L$ condition may not be satisfied, we recently proved in (arXiv:1708.04422) that there exist a non-negative"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.02973","kind":"arxiv","version":2},"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"}