The Chordal Loewner Equation Driven by Brownian Motion with Linear Drift

Persistent Link:
http://hdl.handle.net/10150/195702
Title:
The Chordal Loewner Equation Driven by Brownian Motion with Linear Drift
Author:
Dyhr, Benjamin Nicholas
Issue Date:
2009
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
Keywords:
Brownian motion; Mathematical physics; Schramm-Loewner evolution
Degree Name:
Ph.D.
Degree Level:
doctoral
Degree Program:
Mathematics; Graduate College
Degree Grantor:
University of Arizona
Advisor:
Kennedy, Thomas G.
Committee Chair:
Kennedy, Thomas G.

Full metadata record

DC FieldValue Language
dc.language.isoENen_US
dc.titleThe Chordal Loewner Equation Driven by Brownian Motion with Linear Driften_US
dc.creatorDyhr, Benjamin Nicholasen_US
dc.contributor.authorDyhr, Benjamin Nicholasen_US
dc.date.issued2009en_US
dc.publisherThe University of Arizona.en_US
dc.rightsCopyright © 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.en_US
dc.description.abstractSchramm-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.en_US
dc.typetexten_US
dc.typeElectronic Dissertationen_US
dc.subjectBrownian motionen_US
dc.subjectMathematical physicsen_US
dc.subjectSchramm-Loewner evolutionen_US
thesis.degree.namePh.D.en_US
thesis.degree.leveldoctoralen_US
thesis.degree.disciplineMathematicsen_US
thesis.degree.disciplineGraduate Collegeen_US
thesis.degree.grantorUniversity of Arizonaen_US
dc.contributor.advisorKennedy, Thomas G.en_US
dc.contributor.chairKennedy, Thomas G.en_US
dc.contributor.committeememberFaris, William G.en_US
dc.contributor.committeememberPickrell, Douglas M.en_US
dc.contributor.committeememberStepanov, Mikhailen_US
dc.identifier.proquest10579en_US
dc.identifier.oclc659752323en_US
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