Fractional Fokker-Planck-Kolmogorov equations with Hölder continuous drift

Abstract

We study the fractional Fokker-Planck-Kolmogorov equation with the fractional index \(\alpha \in [1,2)\) and use a vector-valued Calderón-Zygmund theorem to obtain the existence and uniqueness of \(L^p([0,T];{{\mathcal {C}}}_b^{\alpha +\beta }({{\mathbb {R}}}^d))\cap W^{1,p}([0,T];{{\mathcal {C}}}_b^\beta ({{\mathbb {R}}}^d))\) solution under the assumptions that the drift coefficient and nonhomogeneous term are in \(L^p([0,T];{{\mathcal {C}}}_b^{\beta }({{\mathbb {R}}}^d))\) with \(p\in [\alpha /(\alpha -1),+\infty ]\) and \(\beta \in (0,1)\). As applications, we prove the unique strong solvability as well as Davie’s type uniqueness of time inhomogeneous stochastic differential equation with the drift in \(L^p([0,T];{{\mathcal {C}}}_b^{\beta }({\mathbb R}^d;{{\mathbb {R}}}^d))\) and driven by the \(\alpha \)-stable process for \(\beta > 1-\alpha /2\) and \(p>2\alpha /(\alpha +2\beta -2)\).