{"paper":{"title":"Supercritical Superprocesses: Proper Normalization and Non-degenerate Strong Limit","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":"2017-08-15T07:59:19Z","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 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 (see, J. Appl. Probab. 46 (2009), 479--496) that $M_\\infty$ is non-degenerate iff the $L\\log L$ condition is satisfied. In this paper we are mainly interested in the case when the $L\\log L$ condition i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.04422","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"}