Stochastic mechanics is a theory that holds great promise in resolving the mathematical and
interpretational issues encountered in the canonical and path integral formulations of quantum
theories. It provides an equivalent formulation of quantum theories but substantiates it with
a mathematically rigorous stochastic interpretation by means of a stochastic quantization
prescription. The book builds on recent developments in this theory and shows that quantum
mechanics can be unified with the theory of Brownian motion in a single mathematical framework.
Moreover it discusses the extension of the theory to curved spacetime using second order
geometry and the induced Itô deformations of the spacetime symmetries. The book is
self-contained and provides an extensive review of stochastic mechanics of the single spinless
particle. The book builds up the theory on a step by step basis. It starts in chapter 2 with
a review of the classical particle subjected to scalar and vector potentials. In chapter 3 the
theory is extended to the study of a Brownian motion in any potential by the introduction of a
Gaussian noise. In chapter 4 the Gaussian noise is complexified. The result is a complex
diffusion theory that contains both Brownian motion and quantum mechanics as a special limit.
In chapters 5 the theory is extended to relativistic diffusion theories. In chapter 6 the
theory is further generalized to the context of pseudo-Riemannian geometry. Finally in chapter
7 some interpretational aspects of the stochastic theory are discussed in more detail. The
appendices concisely review relevant notions from probability theory stochastic processes
stochastic calculus stochastic differential geometry and stochastic variational calculus. The
book is aimed at graduate students and researchers in theoretical physics and applied
mathematics with an interest in the foundations of quantum theory and Brownian motion. The book
can be used as reference material for courses on and further research in stochastic mechanics
stochastic quantization diffusion theories on curved spacetimes and quantum gravity.
