{"paper":{"title":"Parametrix construction of the transition probability density of the solution to an SDE driven by $\\alpha$-stable noise","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Alexei Kulik, Victoria Knopova","submitted_at":"2014-12-30T19:02:23Z","abstract_excerpt":"Let $L:= -a(x) (-\\Delta)^{\\alpha/2}+ (b(x), \\nabla)$, where $\\alpha\\in (0,2)$, and $a:\\rd\\to (0,\\infty)$, $b: \\rd\\to \\rd$. Under certain regularity assumptions on the coefficients $a$ and $b$, we associate with the $C_\\infty(\\rd)$-closure of $(L, C_\\infty^2(\\rd))$ a Feller Markov process $X$, which possesses a transition probability density $p_t(x,y)$.\n  To construct this transition probability density and to obtain the two-sided estimates on it, we develop a new version of the parametrix method, which allows us to handle the case $0<\\alpha\\leq 1$ and $b\\neq 0$, i.e. when the gradient part of "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.8732","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"}