{"paper":{"title":"Stochastic Differential Equation for Brox Diffusion","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Khoa L\\^e, Leonid Mytnik, Yaozhong Hu","submitted_at":"2015-06-07T16:01:34Z","abstract_excerpt":"This paper studies the weak and strong solutions to the stochastic differential equation $ dX(t)=-\\frac12 \\dot W(X(t))dt+d\\mathcal{B}(t)$, where $(\\mathcal{B}(t), t\\ge 0)$ is a standard Brownian motion and $W(x)$ is a two sided Brownian motion, independent of $\\mathcal{B}$. It is shown that the It\\^o-McKean representation associated with any Brownian motion (independent of $W$) is a weak solution to the above equation. It is also shown that there exists a unique strong solution to the equation. It\\^o calculus for the solution is developed. For dealing with the singularity of drift term $\\int_0"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.02280","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"}