English

On stochastic equations with drift in $L_{d}$

Probability 2020-09-03 v3

Abstract

For It\^o stochastic equations in Rd\mathbb{R}^{d} with drift in LdL_{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/(d1)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 Wp2W^{2}_{p}-theory of fully nonlinear elliptic equations, the fact that Green's functions are in the class AA_{\infty} of Muckenhoupt and a few other results.

Keywords

Cite

@article{arxiv.2001.04008,
  title  = {On stochastic equations with drift in $L_{d}$},
  author = {N. V. Krylov},
  journal= {arXiv preprint arXiv:2001.04008},
  year   = {2020}
}

Comments

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

R2 v1 2026-06-23T13:09:09.259Z