We consider the parabolic Anderson problem \u2202tu = \u03ba\u2206u + \u03beu on (0,\u221e)\u00d7 Zd with random i.i.d. potential \u03be =( \u03be(z))z\u2208Zd and the initial condition u(0,\u00b7) \u2261 1. Our main assumption is that esssup\u03be(0) = 0. In dependence of the thickness of the distribution Prob(\u03be(0) \u2208\u00b7) close to its essential supremum, we identify both the asymptotics of the moments of u(t,0) and the almost-sure asymptotics of u(t,0) as t \u2192\u221ein terms of variational problems. As a by-product, we establish Lifshitz tails for the random Schro\u00a8dinger operator \u2212\u03ba\u2206\u2212 \u03be at the bottom of its spectrum. In our class of \u03be-distributions, the Lifshitz exponent ranges from d 2 to \u221e; the power law is typically accompanied by lower-order corrections.<\/p>\n","protected":false},"excerpt":{"rendered":"
We consider the parabolic Anderson problem \u2202tu = \u03ba\u2206u + \u03beu on (0,\u221e)\u00d7 Zd with random i.i.d. potential \u03be =( \u03be(z))z\u2208Zd and the initial condition u(0,\u00b7) \u2261 1. Our main assumption is that esssup\u03be(0) = 0. In dependence of the thickness of the distribution Prob(\u03be(0) \u2208\u00b7) close to its essential supremum, we identify both 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":[{"type":"text","value":"Marek Biskup"},{"type":"text","value":"Wolfgang Konig"}],"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-2000-36","msr_organization":"","msr_pages_string":"38","msr_page_range_start":"38","msr_page_range_end":"","msr_series":"","msr_volume":"","msr_copyright":"","msr_conference_name":"","msr_doi":"","msr_arxiv_id":"","msr_mag_id":"","msr_other_authors":"Marek Biskup, Wolfgang 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