On stochastic equations with drift in $L_d$

Abstract

For Itô stochastic equations in ${\mathbb{R}}^d$ with drift in $L_d$ several results are discussed such as the existence of weak solutions, the existence of the corresponding Markov process, Aleksandrov type estimates of their Green's functions, which yield their summability to the power of $d/(d-1)$, 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 $W_p^2$-theory of fully nonlinear elliptic equations, the fact that Green's functions are in the class $A_{\infty }$ of Muckenhoupt and a few other results.

31 pages, one remark rewritten, numerous misprints corrected, a reference updated, Theorem 2.6 is corrected

Cited by in corpus (8)

• SDEs with critical time dependent drifts: strong solutions
• SDEs with critical time dependent drifts: weak solutions
• On time inhomogeneous stochastic Itô equations with drift in $L_{d+1}$
• On strong solutions of Itô's equations with a$\in W_d^1$ and b$\in L_d$
• Elliptic equations with VMO a, b$\in L_d$, and c$\in L_{d/2}$
• On stochastic Itô processes with drift in $L_d$
• Linear and fully nonlinear elliptic equations with Morrey drift
• On potentials of Itô's processes with drift in $L_{d+1}$