Singular Time-Homogeneous Ito Equations and PDEs (Mathematical Surveys and Monographs)

The aim of the book is to present some recent results in the theory of stochastic Ito equations with singular deterministic part (drift) and its applications to second-order elliptic and parabolic equations with singular first-order coefficients.The singularity is characterized by means of Morrey spaces, and this allows for much more singular coefficients than those from Lebesgue spaces. The first five chapters deal with equations having just measurable coefficients and treat the Markov diffusion processes $X$ corresponding to elliptic operators.In particular, Aleksandrov estimates, the Harnack inequality and the Holder continuity of $X$-harmonic functions are analyzed.This analysis requires the corresponding results in PDEs such as the extended Aleksandrov maximum principle, the Harnack inequality and the Holder continuity of PDE-harmonic functions. The three remaining chapters are devoted to the study of weak and strong solutions of Ito equations.This requires some regularity restric