Stochastic It\^o Equations and Parabolic Second-Order Equations with singular Drift

The aim of the book is to present some recent results in the theory of stochastic It\^o 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. For instance, first-order coefficients having behavior like $1/|x|$ near the origin are allowed. In the first part of the book we are dealing with equations having just measurable coefficients and treat the Markov diffusion time-inhomogeneous processes $X$ corresponding to parabolic operators. In particular, mixed-norm parabolic Aleksandrov estimates, Harnack inequality and H\"older continuity of $X$-caloric functions are investigated. This produces the corresponding results in PDEs such as extended Aleksandrov maximum principle, Harnack inequality and H\"older continuity of PDE-caloric functions. In two remaining chapters we concentrate on weak and strong solutions of It\^o equations which requires some regularity restrictions on the diffusion matrix (or second-order coefficients in the PDE language). We give the best to date conditions in terms of Morrey spaces for the existence and uniqueness of weak and strong solutions of It\^o equations with singular drift. The majority of our main results are new even if the drift part is zero.