{"paper":{"title":"The Space-Fractional Poisson Process","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Enzo Orsingher, Federico Polito","submitted_at":"2011-07-14T17:31:13Z","abstract_excerpt":"In this paper we introduce the space-fractional Poisson process whose state probabilities $p_k^\\alpha(t)$, $t>0$, $\\alpha \\in (0,1]$, are governed by the equations $(\\mathrm d/\\mathrm dt)p_k(t) = -\\lambda^\\alpha (1-B)p_k^\\alpha(t)$, where $(1-B)^\\alpha$ is the fractional difference operator found in the study of time series analysis. We explicitly obtain the distributions $p_k^\\alpha(t)$, the probability generating functions $G_\\alpha(u,t)$, which are also expressed as distributions of the minimum of i.i.d.\\ uniform random variables. The comparison with the time-fractional Poisson process is i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.2874","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"}