This book is a concise, self-contained and mathematically rigorous introduction into the Nelson stochastic mechanics. The text written very clearly on the graduate level may be strongly recommended to anybody who is interesting in the applications of stochastic methods in quantum mechanics.
In a brief introduction some probabilistic aspects of classical and quantum physics are discussed including Brownian motion, Feynman path integrals, Wigner distributions and Madelung hydrodynamical approach to quantum mechanics. Starting with the Wiener process the general theory of stochastic diffusion processes is outlined. Itô stochastic calculus, kinematics of diffusion, the notion of stochastic acceleration are introduced and illustrated by several examples.
In the next chapter Nelson stochastic dynamics is defined in terms of the stochastic Newton law and its mathematical equivalence to the Madelung representation of quantum mechanics is shown. The authors give also the proof of existence for diffusions with singular drifts with assures the consistency of the presented scheme and derive the stochastic Newton law from variational principles.
In the last but one chapter the following fundamental problems and methods of quantum mechanics are discussed from the point of view of stochastic mechanics: the structure of quantum observables, repeated measurements, indeterminacy relations, scattering theory, spin and statistics, Euclidean quantum mechanics, semiclassical limit. The final chapter is devoted to the nonquantum applications of stochastic mechanics. The different trapping phenomena and formation of spatial patterns may be described in terms of stochastic Newton law with suitable external forces. The formation of the solar system, could covering of the plantes and van Allen radiation belts are examples of such phenomena. The Appendix contains the review of the basic notions of probability theory.