Annual Member
| Author | Nagasawa, Masao. author |
|---|---|
| Title | Schrรถdinger Equations and Diffusion Theory [electronic resource] / by Masao Nagasawa |
| Imprint | Basel : Birkhรคuser Basel : Imprint: Birkhรคuser, 1993 |
| Connect to | http://dx.doi.org/10.1007/978-3-0348-8568-3 |
| Descript | XII, 323 p. online resource |
I Introduction and Motivation -- 1.1 Quantization -- 1.2 Schrรถdinger Equation -- 1.3 Quantum Mechanics and Diffusion Processes -- 1.4 Equivalence of Schrรถdinger and Diffusion Equations -- 1.5 Time Reversal and Duality -- 1.6 QED and Quantum Field Theory -- 1.7 What is the Schrรถdinger 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 Kacโs Semi-Group and its Renormalization -- 2.8 Time Change -- 2.9 Dirichlet Problem -- 2.10 Fellerโs One-Dimensional Diffusion Processes -- 2.11 Fellerโs Test -- III Duality and Time Reversal of Diffusion Processes -- 3.1 Kolmogoroffโs Duality -- 3.2 Time Reversal of Diffusion Processes -- 3.3 Duality of Time-Inhomogeneous Diffusion Processes -- 3.4 Schrรถdingerโs and Kolmogoroff s Representations -- 3.5 Some Remarks -- IV Equivalence of Diffusion and Schrรถdinger Equations -- 4.1 Change of Variable Formulae -- 4.2 Equivalence Theorem -- 4.3 Discussion of the Non-Linear Dependence -- 4.4 A Solution to Schrรถdingerโs 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 Csiszarโs Projection Theorem -- 5.3 Reference Processes -- 5.4 Diffusion Processes in Schrรถdingerโs 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 the q-Representation -- 6.4 What is the Feynman Integral -- 6.5 A Remark on Kacโs 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 Schrรถdinger Equations with Singular Potentials -- VIII The Schrรถdinger 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 Schrรถdinger Equations -- 8.4 Some Comments -- IX Applications of the Statistical Model for Schrรถdinger 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 Csiszarโs Projection -- 10.1 Relative Entropy -- 10.2 Csiszarโs 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 Fรฉnyesโ โEquation of Motionโ 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
