{"paper":{"title":"The parabolic Anderson model in a dynamic random environment: basic properties of the quenched Lyapunov exponent","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Dirk Erhard, Frank den Hollander, Gr\\'egory Maillard","submitted_at":"2012-08-01T19:58:41Z","abstract_excerpt":"In this paper we study the parabolic Anderson equation \\partial u(x,t)/\\partial t=\\kappa\\Delta u(x,t)+\\xi(x,t)u(x,t), x\\in\\Z^d, t\\geq 0, where the u-field and the \\xi-field are \\R-valued, \\kappa \\in [0,\\infty) is the diffusion constant, and $\\Delta$ is the discrete Laplacian. The initial condition u(x,0)=u_0(x), x\\in\\Z^d, is taken to be non-negative and bounded. The solution of the parabolic Anderson equation describes the evolution of a field of particles performing independent simple random walks with binary branching: particles jump at rate 2d\\kappa, split into two at rate \\xi\\vee 0, and di"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.0330","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":""},"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"}