On stochastic equations with drift in Ld

Abstract

For the Itô stochastic equations in ${\mathbb{R}}^{\mathit{d}}$ with drift in ${\mathit{L}}_{\mathit{d}}$, several results are discussed, such as the existence of weak solutions, the existence of the corresponding Markov process, the Aleksandrov type estimates of their Green’s functions, which yield their summability to the power of $\mathit{d}/(\mathit{d}-1)$, the Fabes–Stroock type estimates, which show that Green’s functions are summable to a higher degree, the Fanghua Lin type estimates, which are one of the main tools in the ${\mathit{W}}_{\mathit{p}}^2$-theory of fully nonlinear elliptic equations, the fact that Green’s functions are in the class ${\mathit{A}}_{\infty }$ of Muckenhoupt and a few other results.