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

TL;DR

This book explores recent advances in stochastic Itô equations with highly singular drifts, applying Morrey space analysis to establish existence, uniqueness, and regularity results for solutions and related PDEs.

Contribution

It introduces new conditions in Morrey spaces for the existence and uniqueness of solutions to Itô equations with singular drifts, extending previous results.

Findings

Established Harnack inequalities and Hölder continuity for associated PDEs.

Derived new conditions for solution existence and uniqueness in Morrey spaces.

Extended classical PDE results to equations with more singular coefficients.

Abstract

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.