Mathematical and physical aspects of stochastic mechanics / Ph. Blanchard, Ph. Combe, W. Zeng

This slim volume contains lecture notes from a summer graduate course given in Bielefeld. The exposition has plainly been extensively worked over (thirty-two people, amongst whom are leaders in the field, are thanked for advice and encouragement). The result, while far from giving all the details necessary in a rigorous development of the subject matter, provides a well reasoned and motivated introduction from a modern perspective.
An introductory chapter discusses some of the early work, and in particular the Hamilton-Jacobi and Madelung fluids. The second chapter, on kinematics of stochastic diffusion processes, develops material from the theory of stochastic processes that will be required later. Chapter III is on Nelson's form of stochastic mechanics, with the existence of Nelson's diffusion processes discussed in the following chapter. Chapter V discusses stochastic variational principles following Yasue's line, but with mention of other approaches. Chapter VI is entitled "Two viewpoints concerning quantum and stochastic mechanics''. The following chapter "A nonquantal look at stochastic mechanics'' mentions trapping phenomena and spatial patterns, jet-streams in nebulae, clouds on planets, the Van Allen radiation belt, and transport in plasma, all inside eight pages; the result is not much of an exposition of any of the topics. Finally there are some mathematical appendices and an extensive bibliography of 116 items. (MathSciNet)