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Title: Topics on stochastic Schrödinger equations and estimates for the derivative of diffusion semigroups
Author: Rincon Solis, L. A.
Awarding Body: University of Wales Swansea
Current Institution: Swansea University
Date of Award: 1999
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This thesis treats some topics in the overlap of classical probability theory and non-relativistic quantum mechanics. The main contributions of the present work are: 1. A proof of a path integral representation for the solutions of certain types of stochastic Schrödinger equations driven by semimartingales (Chapter 3). 2. Explicit calculations of the kernels (propagators) for stochastic quantum harmonic oscillators (Chapter 4). Our formulae are stochastic extensions of the classical Mehler kernel formula. 3. A proof of the equivalence between shock waves for the Burgers' equation for vanishing viscosity, and caustics for the Schrödinger equation in the classical limit (Chapter 2). 4. Formulae for upper bound estimates for the first derivative of diffusion semigroups for small time (Chapter 5). The thesis is in three parts: In Part I (Chapters 1 and 2), we apply some results of Part II regarding the appearance of caustics in the classical limit of quantum mechanics. We also give a short account of some stochastic approaches to quantum mechanics, and a brief expository account of the connection between ray optics and the classical limit of quantum mechanics. In Part II (Chapters 3 and 4), we study path integral solutions of stochastic Schrödinger equations, and derive our stochastic Mehler kernel formulae. In Part III (Chapter 5), we study some perturbations of the infintesimal generator of a diffusion semigroup, and find estimates for the derivative of the corresponding semigroups for small values of the time parameter.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available