Schrödinger Equations and Diffusion Theory addresses the question
"What is the Schrödinger equation?" in terms of diffusion processes, and
shows that the Schrödinger equation and diffusion equations in duality
are equivalent. In turn, Schrödinger's conjecture of 1931 is solved. The
theory of diffusion processes for the Schrödinger equation tells us that
we must go further into the theory of systems of (infinitely) many
interacting quantum (diffusion) particles.
The method of relative entropy and the theory of transformations enable
us to construct severely singular diffusion processes which appear to be
equivalent to Schrödinger equations.
The theory of large deviations and the propagation of chaos of
interacting diffusion particles reveal the statistical mechanical nature
of the Schrödinger equation, namely, quantum mechanics.
The text is practically self-contained and requires only an elementary
knowledge of probability theory at the graduate level.
*This book is a self-contained, very well-organized monograph
recommended to researchers and graduate students in the field of
probability theory, functional analysis and quantum dynamics. (...) what
is written in this book may be regarded as an introduction to the theory
of diffusion processes and applications written with the physicists in
mind. Interesting topics present themselves as the chapters proceed.
(...) this book is an excellent addition to the literature of
mathematical sciences with a flavour different from an ordinary textbook
in probability theory because of the author's great contributions in
this direction. Readers will certainly enjoy the topics and appreciate
the profound mathematical properties of diffusion processes.
*(Mathematical Reviews)
