{"paper":{"title":"$L^q(L^p)$-theory of stochastic differential equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Guohuan Zhao, Longjie Xie, Pengcheng Xia, Xicheng Zhang","submitted_at":"2019-08-04T01:52:13Z","abstract_excerpt":"In this paper we show the weak differentiability of the unique strong solution with respect to the starting point $x$ as well as Bismut-Elworthy-Li's derivative formula for the following stochastic differential equation in $\\mathbb R^d$: $$ {\\rm d} X_t=b(t,X_t){\\rm d} t+\\sigma(t,X_t){\\rm d} W_t,\\ \\ X_0=x\\in\\mathbb R^d, $$ where $\\sigma$ is bounded, uniformly continuous and nondegenerate, $\\nabla\\sigma\\in \\widetilde{\\mathbb L}^{p_1}_{q_1}$ and $b\\in \\widetilde{\\mathbb L}^{p_2}_{q_2}$ for some $p_i,q_i\\in[2,\\infty)$ with $\\frac{d}{p_i}+\\frac{2}{q_i}<1$, $i=1,2$, where $\\widetilde{\\mathbb L}^{p_i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.01255","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1908.01255/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"}