{"paper":{"title":"Biased tug-of-war, the biased infinity Laplacian, and comparison with exponential cones","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OC","math.PR"],"primary_cat":"math.AP","authors_text":"G\\'abor Pete, Stephanie Somersille, Yuval Peres","submitted_at":"2008-11-03T02:51:51Z","abstract_excerpt":"We prove that if U\\subset\\R^n is an open domain whose closure \\overline{U} is compact in the path metric, and F is a Lipschitz function on \\partial{U}, then for each \\beta\\in\\R there exists a unique viscosity solution to the \\beta-biased infinity Laplacian equation \\beta |\\nabla u| + \\Delta_\\infty u=0 on U that extends F, where \\Delta_\\infty u= |\\nabla u|^{-2} \\sum_{i,j} u_{x_i}u_{x_ix_j} u_{x_j}.\n  In the proof, we extend the tug-of-war ideas of Peres, Schramm, Sheffield and Wilson, and define the \\beta-biased \\eps-game as follows. The starting position is x_0 \\in U. At the k^\\text{th} step t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0811.0208","kind":"arxiv","version":3},"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"}