This monograph has arisen out of a number of attempts spanning almost five decades to understand how one might examine the evolution of densities in systems whose dynamics are described by differential delay equations. Though the authors have no definitive solution to the problem, they offer this contribution in an attempt to define the problem as they see it, and to sketch out several obvious attempts that have been suggested to solve the problem and which seem to have failed. They hope that by being available to the general mathematical community, they will inspire others to considerand hopefully solvethe problem. Serious attempts have been made by all of the authors over the years and they have made reference to these where appropriate.

Describes an open and difficult mathematical problem, offering a new research field for young investigators Gives a brief but unified contrasting description of examination of dynamics from a trajectory or an ensemble point of view Highlights all of the problems attendant to the development of an appropriate measure to examine ergodic behavior in infinite-dimensional dynamical systems Presents possible applications of functional calculus to infinite dimensional dynamical systems Offers a motivated area of research combining different fields of mathematics

Zusammenfassung

"The book is devoted to the problem of evolution of densities in dynamical systems described by delay differential equations. ... The book is interesting and introduces the reader to the problem of the evolution of the densities. Indeed, formulating (if possible) such an evolution one is able to guess the asymptotic behavior of the solutions with known density of their initial values. The examples in the book are carefully selected in order to illustrate the main theoretical results." (George Karakostas, zbMATH 1462.37001, 2021)

Inhalt
Part I. Introduction and Background to Density Evolution Problems.- 1. Introduction and Motivation.- 2. Density Evolution in Systems with Finite Dimensional Dynamics.- Part II. Illustrating the Problem and Making it Precise for Differential Delay Equations.- 3. Dynamics in Ensembles of Differential Delay Equations.- 4. The Problem.- III. Possible Analytical Approaches.- 5. The Hopf Functional Approach.- 6. The Method of Steps.- Part IV. Possible Approximating Solutions.- 7. Turning a Differential Delay Equation into a High-Dimensional Map.- 8. Approximate "Liouville-like" Equation.- 9. Summary and Conclusions.- References.- Index.