Schrödinger Equations and Diffusion Theory

Addressing the question "what is the Schroedinger equation" in terms of diffusion processes, this work shows that the Schroedinger equation and diffusion equations in duality are equivalent. In turn, Schroedinger's conjecture of 1931 is solved. The theory of diffusion processes for the Schroedinger 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 Schroedinger equations. The theory of large deviations and the propagation of chaos of interacting diffusion particles reveal the statistical mechanical nature of the Schroedinger equation, namely, quantum mechanics. The text is practically self-contained and requires only an elementary knowledge of probability theory at the graduate level.