{"paper":{"title":"Spatial Brownian motion in renormalized Poisson potential: A critical case","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Jan Rosinski, Xia Chen","submitted_at":"2011-03-29T17:37:07Z","abstract_excerpt":"Let $B_s$ be a three dimensional Brownian motion and $\\omega(dx)$ be an independent Poisson field on $\\mathbb{R}^3$. It is proved that for any $t>0$, conditionally on $\\omega(\\cdot)$,\n  \\label{*} \\mathbb{E}_0 \\exp\\{\\theta \\int_0^t \\bar{V}(B_s) ds\\} \\  < \\infty \\ a.s. & \\text{if} \\theta< 1/16, \\medskip\n  = \\infty \\ a.s. & \\text{if} \\theta> 1/16, where $\\bar{V}(x)$ is the renormalized Poisson potential\n  $$ \\bar{V}(x)=\\int_{\\mathbb{R}^3} \\frac{1}{| x-y |^2} \\big[\\omega(dy)-dy\\big]. $$ Then the long term behavior of the quenched exponential moment \\eqref{*} is determined for $\\theta \\in (0, 1/16)"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.5717","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"}