{"paper":{"title":"Fractal Geometry of the Valleys of the Parabolic Anderson Equation","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.PR","authors_text":"Jaeyun Yi, Promit Ghosal","submitted_at":"2021-08-09T04:59:36Z","abstract_excerpt":"We study the macroscopic fractal properties of the deep valleys of the solution of the $(1+1)$-dimensional parabolic Anderson equation $${\\partial \\over \\partial t}u(t,x) =\\frac{1}{2} {\\partial^2 \\over \\partial x^2} u(t,x) + u(t,x)\\dot{W}(t,x),t>0, x\\in {\\bf R},\\quad\n  u(0,x) \\equiv u_0(x),x\\in {\\bf R}, $$ where $\\dot{W}$ is the time-space white noise and $0<\\inf_{x\\in {\\bf R}} u_0(x)\\leq \\sup_{x\\in {\\bf R}} u_0(x)<\\infty.$ Unlike the macroscopic multifractality of the tall peaks, we show that valleys of the parabolic Anderson equation are macroscopically monofractal. In fact, the macroscopic "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2108.03810","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2108.03810/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}