{"paper":{"title":"Feynman-Kac representation for the parabolic Anderson model driven by fractional noise","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.PR","authors_text":"Kamran Kalbasi, Thomas S. Mountford","submitted_at":"2017-06-27T21:16:05Z","abstract_excerpt":"We consider the parabolic Anderson model driven by fractional noise: $$ \\frac{\\partial}{\\partial t}u(t,x)= \\kappa \\boldsymbol{\\Delta} u(t,x)+ u(t,x)\\frac{\\partial}{\\partial t}W(t,x) \\qquad x\\in\\mathbb{Z}^d\\;,\\; t\\geq 0\\,, $$ where $\\kappa>0$ is a diffusion constant, $\\boldsymbol{\\Delta}$ is the discrete Laplacian defined by $\\boldsymbol{\\Delta} f(x)= \\frac{1}{2d}\\sum_{|y-x|=1}\\bigl(f(y)-f(x)\\bigr)$, and $\\{W(t,x)\\;;\\;t\\geq0\\}_{x \\in \\mathbb{Z}^d}$ is a family of independent fractional Brownian motions with Hurst parameter $H\\in(0,1)$, indexed by $\\mathbb{Z}^d$. We make sense of this equation v"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.09050","kind":"arxiv","version":1},"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"}