{"id":152470,"date":"2000-06-01T00:00:00","date_gmt":"2000-06-01T00:00:00","guid":{"rendered":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/msr-research-item\/screening-effect-due-to-heavy-lower-tails-on-one-dimensional-parabolic-anderson-model\/"},"modified":"2018-10-16T19:59:15","modified_gmt":"2018-10-17T02:59:15","slug":"screening-effect-due-to-heavy-lower-tails-on-one-dimensional-parabolic-anderson-model","status":"publish","type":"msr-research-item","link":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/publication\/screening-effect-due-to-heavy-lower-tails-on-one-dimensional-parabolic-anderson-model\/","title":{"rendered":"Screening Effect Due to Heavy Lower Tails on One-dimensional Parabolic Anderson Model"},"content":{"rendered":"

We consider the large-time behavior of the solution u<\/mi>:<\/mo>[<\/mo>0<\/mn>,<\/mo>∞<\/mi>)<\/mo>×<\/mo>Z<\/mi><\/mrow>→<\/mo>[<\/mo>0<\/mn>,<\/mo>∞<\/mi>)<\/mo><\/math>\">u<\/span>:<\/span>[<\/span>0<\/span>,<\/span>\u221e<\/span>)<\/span>\u00d7<\/span>Z<\/span><\/span><\/span>\u2192<\/span>[<\/span>0<\/span>,<\/span>\u221e<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span> to the parabolic Anderson problem \u2202t<\/sub>u=\u03ba\u0394u+\u03beu with initial data u(0,\u00a0\u00b7)=1 and non-positive finite i.i.d. potentials (<\/mo>ξ<\/mi>(<\/mo>z<\/mi>)<\/mo>)<\/mo>z<\/mi>∈<\/mo>Z<\/mi><\/mrow><\/mrow><\/msub><\/math>\">(<\/span>\u03be<\/span>(<\/span>z<\/span>)<\/span>)<\/span>z<\/span>\u2208<\/span>Z<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>. Unlike in dimensions d\u22652, the almost-sure decay rate of u(t,\u00a00) as t\u2192\u221e is not determined solely by the upper tails of \u03be(0); too heavy lower tails of \u03be(0) accelerate the decay. The interpretation is that sites x with large negative \u03be(x) hamper the mass flow and hence screen off the influence of more favorable regions of the potential. The phenomenon is unique to d=1. The result answers an open question from our previous study [BK00] of this model in general dimension.<\/p>\n","protected":false},"excerpt":{"rendered":"

We consider the large-time behavior of the solution u:[0,\u221e)\u00d7Z\u2192[0,\u221e) to the parabolic Anderson problem \u2202tu=\u03ba\u0394u+\u03beu with initial data u(0,\u00a0\u00b7)=1 and non-positive finite i.i.d. potentials (\u03be(z))z\u2208Z. Unlike in dimensions d\u22652, the almost-sure decay rate of u(t,\u00a00) as t\u2192\u221e is not determined solely by the upper tails of \u03be(0); too heavy lower tails of \u03be(0) accelerate the […]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr-author-ordering":null,"msr_publishername":"","msr_publisher_other":"","msr_booktitle":"","msr_chapter":"","msr_edition":"","msr_editors":"","msr_how_published":"","msr_isbn":"","msr_issue":"","msr_journal":"","msr_number":"MSR-TR-2000-76","msr_organization":"","msr_pages_string":"13","msr_page_range_start":"13","msr_page_range_end":"","msr_series":"","msr_volume":"","msr_copyright":"","msr_conference_name":"","msr_doi":"10.1023\/A:1004840328675","msr_arxiv_id":"","msr_s2_paper_id":"","msr_mag_id":"","msr_pubmed_id":"","msr_other_authors":"Marek Biskup, Wolfgang 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