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    The Chordal Loewner Equation Driven by Brownian Motion with Linear Drift

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    Author
    Dyhr, Benjamin Nicholas
    Issue Date
    2009
    Keywords
    Brownian motion
    Mathematical physics
    Schramm-Loewner evolution
    Advisor
    Kennedy, Thomas G.
    Committee Chair
    Kennedy, Thomas G.
    
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    Publisher
    The University of Arizona.
    Rights
    Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.
    Abstract
    Schramm-Loewner evolution (SLE(kappa)) is an important contemporary tool for identifying critical scaling limits of two-dimensional statistical systems. The SLE(kappa) one-parameter family of processes can be viewed as a special case of a more general, two-parameter family of processes we denote SLE(kappa, mu). The SLE(kappa, mu) process is defined for kappa>0 and real numbers mu; it represents the solution of the chordal Loewner equations under special conditions on the driving function parameter which require that it is a Brownian motion with drift mu and variance kappa. We derive properties of this process by use of methods applied to SLE(kappa) and application of Girsanov's Theorem. In contrast to SLE(kappa), we identify stationary asymptotic behavior of SLE(kappa, mu). For kappa in (0,4] and mu > 0, we present a pathwise construction of a process with stationary temporal increments and stationary imaginary component and relate it to the limiting behavior of the SLE(kappa, mu) generating curve. Our main result is a spatial invariance property of this process achieved by defining a top-crossing probability for points in the upper half plane with respect to the generating curve.
    Type
    text
    Electronic Dissertation
    Degree Name
    Ph.D.
    Degree Level
    doctoral
    Degree Program
    Mathematics
    Graduate College
    Degree Grantor
    University of Arizona
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