Corresponding to the link of Itô's stochastic differential equations (SDEs) and linear parabolic equations, distribution dependent SDEs (DDSDEs) characterize nonlinear Fokker–Planck equations. This type of SDEs is named after McKean–Vlasov due to the pioneering work of H P McKean (1966), where an expectation dependent SDE is proposed to characterize nonlinear PDEs for Maxwellian gas. Moreover, by using the propagation of chaos for Kac particle systems, weak solutions of DDSDEs are constructed as weak limits of mean field particle systems when the number of particles goes to infinity, so that DDSDEs are also called mean-field SDEs. To restrict a DDSDE in a domain, we consider the reflection boundary by following the line of A V Skorohod (1961).

This book provides a self-contained account on singular SDEs and DDSDEs with or without reflection. It covers well-posedness and regularities for singular stochastic differential equations; well-posedness for singular reflected SDEs; well-posedness of singular DDSDEs; Harnack inequalities and derivative formulas for singular DDSDEs; long time behaviors for DDSDEs; DDSDEs with reflecting boundary; and killed DDSDEs.

Contents:

• Singular Stochastic Differential Equations


• Singular Reflected SDEs


• DDSDEs: Well-posedness


• DDSDEs: Harnack Inequality and Derivative Estimates


• DDSDEs: Long Time Behaviors


• DDSDEs with Reflecting Boundary


• Killed DDSDEs

Readership: Postgraduate students and researchers in the areas of stochastic analysis, stochastic differential equations, stochastic partial differential equations, and stochastic dynamics. The book may be used as textbook for advanced courses on stochastic analysis.

Key Features:

• Distribution dependent stochastic differential equation is a very active research topic in the frontier of stochastic analysis and applications


• Most results introduced in the book are derived in the recent two years, where those on singular reflecting SDEs/DDSDEs first appeared in 2021


• Novelty and advancement are two features of the book, which should be useful for researchers to work further in related directions