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Title: Topics in stochastic analysis
Author: Davies, M. J.
Awarding Body: University College of Swansea
Current Institution: Swansea University
Date of Award: 1993
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This thesis uses Nelson's stochastic mechanics to study a variety of problems. These include point sources, particles in a constant magnetic field with oscillator potentials and Brownian Motion with a constant drift. By using a finite difference approximation Chapter 1 gives an account of the Stochastic Variational Principle for stochastic mechanics based on Carlen's approach. It is shown that every diffusion satisfying the dynamical law of stochastic mechanics corresponds to a solution of the Schröinger equation. Chapter 2 is concerned with point sources and gives a brief account of Nelson's work in this field as well as examples of monochromatic one and two point particle sources as elucidated by Truman et al. The generator of the radial motion for a particle emitted by a point source is shown to be the generator of Brownian Motion with a constant drift. The transition density and expected first hitting times for this process are then derived explicitly. Chapter 3 gives a resuméof Shucker's result for sample paths of the Nelson stochastic process governed by the free wave function. Analogues of Shucker's result for the initial Gaussian wave function in the presence of a constant magnetic field together with positive or negative harmonic oscillator potentials are then proved. Finally Chapter 4 deals with the case of one dimensional Brownian Motion with a constant drift k on the half line (0,∞) with 0 accessible and ∞ inaccessible. Following some earlier work of Mandl the transition density for such a process is obtained explicitly for the most general boundary conditions leading to continuous sample paths and a contraction semigroup on C(0,∞) with generatorhskip 1.5cm. Usingvarious expectation values we obtain the distribution of first hitting times and last exit times for these processes.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available