Schrödinger equations and diffusion theory. (English) Zbl 0780.60003

Monographs in Mathematics. 86. Basel: Birkhäuser Verlag. xii, 320 p. (1993).
This monograph is devoted to an attempt to give an answer to the fundamental and long standing question: “What is the Schrödinger equation?” An exposition is based on the diffusion theory, more precisely on the theory of diffusion processes. It is shown (see Chapter 4) that the Schrödinger equation and diffusion equations in duality are equivalent. This equivalence implies that, roughly speaking, if we add (Brownian) noise and specific additional drift to a “classical particle”, then we get movement of a quantum particle. It means that the quantum particle (=diffusion particle) has its well-defined position but no velocity, and hence it is not at all a “classical particle”.
This book may be regarded as an introduction to the theory of diffusion processes with applications. The Chapters 2, 3, 5 and 6 contain the fundamental review of this theory. In the Chapters 7 and 8 the author discusses some problems of the statistical mechanics. Some applications in biology and physics are given in Chapter 9. Chapters 10 and 11 offer a self-contained exposition on relative entropy and large deviations. Nonlinearity induced by the branching property is briefly explained in Chapter 12.


60-02 Research exposition (monographs, survey articles) pertaining to probability theory
81-02 Research exposition (monographs, survey articles) pertaining to quantum theory