Singular time-homogeneous Itô equations and PDEs / N.V. Krylov

Format:

Book

Author/Creator:

Krylov, N. V. (Nikolaĭ Vladimirovich), author.

Contributor:

American Mathematical Society, issuing body.

Series:

Mathematical surveys and monographs ; 2331-7159 v. 298.

Mathematical surveys and monographs, 2331-7159 ; volume 298

Language:

English

Subjects (All):

Stochastic differential equations.

Stochastic processes.

Singularities (Mathematics).

Stochastic Processes.

Medical Subjects:

Stochastic Processes.

Physical Description:

1 online resource

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2026]

Summary:

"The aim of the book is to present some recent results in the theory of stochastic Itô 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 corresponding to elliptic operators. In particular, Aleksandrov estimates, the Harnack inequality and the Hölder continuity of -harmonic functions are analyzed. This analysis requires the corresponding results in PDEs such as the extended Aleksandrov maximum principle, the Harnack inequality and the Hölder continuity of PDE-harmonic functions. The three remaining chapters are devoted to the study of weak and strong solutions of Itô equations. This requires some regularity restrictions on the diffusion matrix (or second-order coefficients in the PDE language). The book provides the best to date conditions in terms of Morrey spaces for the existence and uniqueness of weak and strong solutions of Itô equations with singular drift. The majority of the main results in the book are new even if the drift part is zero"-- AMS Digital Library

Contents:

Preliminaries

Finer properties

Itô's equations and Markov processes

Further properties of M(d,N,r,p,b)-processes. Itô's formula for u∈W1,2/p,q

Morrey and local Morrey b

Weak solutions of stochastic equation via Morrey spaces

Strong solutions

Weak and strong solutions of stochastic equation via Sobolev spaces

Notes:

Includes bibliographical references and index

Online resource; title from PDF title page (AMS Digital Library, viewed July 22, 2026)

Other Format:

Print version: Krylov, N. V. (Nikolaĭ Vladimirovich) Singular time-homogeneous Itô equations and PDEs

ISBN:

9781470486495

1470486490

OCLC:

1606238634

Access Restriction:

Restricted for use by site license