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This invaluable research monograph presents a unified and fascinating theory of generalized functionals of Brownian motion and other fundamental processes such as fractional Brownian motion and Levy process - covering the classical Wiener-Ito class including the generalized functionals of Hida as special cases, among others. It presents a thorough and comprehensive treatment of the Wiener-Sobolev spaces and their duals, as well as Malliavin calculus with their applications. The presentation is lucid and logical, and is based on a solid foundation of analysis and topology. The monograph develop

Brownian motion processes. --- Stochastic processes. --- Random processes --- Probabilities --- Wiener processes --- Brownian movements --- Fluctuations (Physics) --- Markov processes

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This book offers the first comprehensive presentation of measure-valued solutions for nonlinear deterministic and stochastic evolution equations on infinite dimensional Banach spaces. Unlike traditional solutions, measure-valued solutions allow for a much broader class of abstract evolution equations to be addressed, providing a broader approach. The book presents extensive results on the existence of measure-valued solutions for differential equations that have no solutions in the usual sense. It covers a range of topics, including evolution equations with continuous/discontinuous vector fields, neutral evolution equations subject to vector measures as impulsive forces, stochastic evolution equations, and optimal control of evolution equations. The optimal control problems considered cover the existence of solutions, necessary conditions of optimality, and more, significantly complementing the existing literature. This book will be of great interest to researchers in functional analysis, partial differential equations, dynamic systems and their optimal control, and their applications, advancing previous research and providing a foundation for further exploration of the field.

Differential equations. --- System theory. --- Control theory. --- Functional analysis. --- Mathematics. --- Engineering mathematics. --- Differential Equations. --- Systems Theory, Control . --- Functional Analysis. --- Applications of Mathematics. --- Engineering Mathematics. --- Engineering --- Engineering analysis --- Mathematical analysis --- Math --- Science --- Systems, Theory of --- Systems science --- 517.91 Differential equations --- Differential equations --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Dynamics --- Machine theory --- Mathematics --- Philosophy --- Evolution equations, Nonlinear. --- Banach spaces. --- Equacions d'evolució no lineal --- Espais de Banach

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This book offers the first comprehensive presentation of measure-valued solutions for nonlinear deterministic and stochastic evolution equations on infinite dimensional Banach spaces. Unlike traditional solutions, measure-valued solutions allow for a much broader class of abstract evolution equations to be addressed, providing a broader approach. The book presents extensive results on the existence of measure-valued solutions for differential equations that have no solutions in the usual sense. It covers a range of topics, including evolution equations with continuous/discontinuous vector fields, neutral evolution equations subject to vector measures as impulsive forces, stochastic evolution equations, and optimal control of evolution equations. The optimal control problems considered cover the existence of solutions, necessary conditions of optimality, and more, significantly complementing the existing literature. This book will be of great interest to researchers in functional analysis, partial differential equations, dynamic systems and their optimal control, and their applications, advancing previous research and providing a foundation for further exploration of the field.

Functional analysis --- Differential equations --- Mathematics --- Applied physical engineering --- differentiaalvergelijkingen --- toegepaste wiskunde --- functies (wiskunde) --- economie --- systeemtheorie --- wiskunde

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Mathematics