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We prove existence and uniqueness of strong solutions to stochastic equations in domains G ⊂ R d with unit diffusion and singular time dependent drift b up to an explosion time. We only assume local L q L p -integrability of b in R × G with d/p + 2/q < 1. We also prove strong Feller properties in this case. If b is the gradient in x of a nonnegative function ψ blowing up as G x → ∂G, we prove that the conditions 2D t ψ ≤ Kψ, 2D t ψ + ψ ≤ Ke εψ , ε ∈ [0, 2), imply that the explosion time is infinite and the distributions of the solution have sub Gaussian tails.

Abstract. An L p -theory of divergence and non-divergence form elliptic and parabolic equations is presented. The main coefficients are supposed to belong to the class V M O x , which, in particular, contains all functions independent of x. Weak uniqueness of the martingale problem associated with such equations is obtained.

Given strong uniqueness for an It6's stochastic equation with discontinuous coefficients, we prove that its solution can be constructed on "any" probability space by using, for example, Euler's polygonal approximations. Stochastic equations in IR d and in domains in IR d are considered.

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Mathematics Subject Classification. Primary 35-02, 35J15, 35K10, 60J35, 60J60. For additional information and updates on this book, visit www.ams.org/bookpages/surv-207 Library of Congress Cataloging-in-Publication Data Fokker-Planck-Kolmogorov equations /Vladimir I. Bogachev, Nicolai V. Krylov, Michael Röckner, Stanislav V. Shaposhnikov. pages cm. -(Mathematical surveys and monographs ; volume 207) Includes bibliographical references and index. ISBN 978-1-4704-2558-6 (alk. paper) 1. Fokker-Planck equation. 2. Stochastic differential equations. I. Bogachev, V. I. (Vladimir Igorevich), 1961-II. Krylov, N. V. (Nicolai Vladimirovich). III. Röckner, Michael, 1956-IV. Shaposhnikov, Stanislav V. QA274.23.F65 2015 515 .353-dc23 2015024922Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given.