Schrödinger Equations and Diffusion Theory

I Introduction and Motivation.- 1.1 Quantization.- 1.2 Schrödinger Equation.- 1.3 Quantum Mechanics and Diffusion Processes.- 1.4 Equivalence of Schrödinger and Diffusion Equations.- 1.5 Time Reversal and Duality.- 1.6 QED and Quantum Field Theory.- 1.7 What is the Schrödinger Equation.- 1.8 Mathematical Contents.- II Diffusion Processes and their Transformations.- 2.1 Time Homogeneous Diffusion Processes.- 2.2 Time Inhomogeneous Diffusion Processes.- 2.3 Brownian Motions.- 2.4 Stochastic Differential Equations.- 2.5 Transformation by a Multiplicative Functional.- 2.6 Feynman-Kac Formula.- 2.7 Kac’s Semi-Group and its Renormalization.- 2.8 Time Change.- 2.9 Dirichlet Problem.- 2.10 Feller’s One-Dimensional Diffusion Processes.- 2.11 Feller’s Test.- III Duality and Time Reversal of Diffusion Processes.- 3.1 Kolmogoroff’s Duality.- 3.2 Time Reversal of Diffusion Processes.- 3.3 Duality of Time-Inhomogeneous Diffusion Processes.- 3.4 Schrödinger’s and Kolmogoroff s Representations.- 3.5 Some Remarks.- IV Equivalence of Diffusion and Schrödinger Equations.- 4.1 Change of Variable Formulae.- 4.2 Equivalence Theorem.- 4.3 Discussion of the Non-Linear Dependence.- 4.4 A Solution to Schrödinger’s Conjecture.- 4.5 A Unified Theory.- 4.6 On Quantization.- 4.7 As a Diffusion Theory.- 4.8 Principle of Superposition.- 4.9 Remarks.- V Variational Principle.- 5.1 Problem Setting in p-Representation.- 5.2 Csiszar’s Projection Theorem.- 5.3 Reference Processes.- 5.4 Diffusion Processes in Schrödinger’s Representation.- 5.5 Weak Fundamental Solutions.- 5.6 An Entropy Characterization of the Markov Property.- 5.6 Remarks.- VI Diffusion Processes in q-Representation.- 6.1 A Multiplicative Functional.- 6.2 Flows of Distribution Densities.- 6.3 Discussions on theq-Representation.- 6.4 What is the Feynman Integral.- 6.5 A Remark on Kac’s Semi-Group.- 6.6 A Typical Case.- 6.7 Hydrogen Atom.- 6.8 A Remark on {?a,?b}.- VII Segregation of a Population.- 7.1 Introduction.- 7.2 Harmonic Oscillator.- 7.3 Segregation of a Finite-System of Particles.- 7.4 A Formulation of the Propagation of Chaos.- 7.5 The Propagation of Chaos.- 7.6 Skorokhod Problem with Singular Drift.- 7.7 A Limit Theorem.- 7.8 A Proof of Theorem 7.1.- 7.9 Schrödinger Equations with Singular Potentials.- VIII The Schrödinger Equation can be a Boltzmann Equation.- 8.1 Large Deviations.- 8.2 The Propagation of Chaos in Terms of Large Deviations.- 8.3 Statistical Mechanics for Schrödinger Equations.- 8.4 Some Comments.- IX Applications of the Statistical Model for Schrödinger Equation.- 9.1 Segregation of a Monkey Population.- 9.2 An Eigenvalue Problem.- 9.3 Septation of Escherichia Coli.- 9.4 The Mass Spectrum of Mesons.- 9.5 Titius-Bode Law.- X Relative Entropy and Csiszar’s Projection.- 10.1 Relative Entropy.- 10.2 Csiszar’s Projection.- 10.3 Exponential Families and Marginal Distributions.- XI Large Deviations.- 11.1 Lemmas.- 11.2 Large Deviations of Empirical Distributions.- XII Non-Linearity Induced by the Branching Property.- 12.1 Branching Property.- 12.2 Non-Linear Equations of Branching Processes.- 12.3 Quasi-Linear Parabolic Equations.- 12.4 Branching Markov Processes with Non-Linear Drift.- 12.5 Revival of a Markov Process.- 12.6 Construction of Branching Markov Processes.- Appendix:.- a.1 Fényes’ “Equation of Motion” of Probability Densities.- a.2 Stochastic Mechanics.- a.3 Segregation of a Population.- a.4 Euclidean Quantum Mechanics.- a.5 Remarks.- a.6 Bohmian Mechanics.- References.