Geometry's Fundamental Role in the Stability of Stochastic Differential Equations

Persistent Link:
http://hdl.handle.net/10150/145150
Title:
Geometry's Fundamental Role in the Stability of Stochastic Differential Equations
Author:
Herzog, David Paul
Issue Date:
2011
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:
We study dynamical systems in the complex plane under the effect of constant noise. We show for a wide class of polynomial equations that the ergodic property is valid in the associated stochastic perturbation if and only if the noise added is in the direction transversal to all unstable trajectories of the deterministic system. This has the interpretation that noise in the "right" direction prevents the process from being unstable: a fundamental, but not well-understood, geometric principle which seems to underlie many other similar equations. The result is proven by using Lyapunov functions and geometric control theory.
Type:
Electronic Dissertation; text
Keywords:
Control Theory; Ergodic Property; Invariant Measures; Lyapunov Functions; Stochastic Differential Equations
Degree Name:
Ph.D.
Degree Level:
doctoral
Degree Program:
Graduate College; Mathematics
Degree Grantor:
University of Arizona
Advisor:
Wehr, Jan

Full metadata record

DC FieldValue Language
dc.language.isoenen_US
dc.titleGeometry's Fundamental Role in the Stability of Stochastic Differential Equationsen_US
dc.creatorHerzog, David Paulen_US
dc.contributor.authorHerzog, David Paulen_US
dc.date.issued2011-
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.abstractWe study dynamical systems in the complex plane under the effect of constant noise. We show for a wide class of polynomial equations that the ergodic property is valid in the associated stochastic perturbation if and only if the noise added is in the direction transversal to all unstable trajectories of the deterministic system. This has the interpretation that noise in the "right" direction prevents the process from being unstable: a fundamental, but not well-understood, geometric principle which seems to underlie many other similar equations. The result is proven by using Lyapunov functions and geometric control theory.en_US
dc.typeElectronic Dissertationen_US
dc.typetexten_US
dc.subjectControl Theoryen_US
dc.subjectErgodic Propertyen_US
dc.subjectInvariant Measuresen_US
dc.subjectLyapunov Functionsen_US
dc.subjectStochastic Differential Equationsen_US
thesis.degree.namePh.D.en_US
thesis.degree.leveldoctoralen_US
thesis.degree.disciplineGraduate Collegeen_US
thesis.degree.disciplineMathematicsen_US
thesis.degree.grantorUniversity of Arizonaen_US
dc.contributor.advisorWehr, Janen_US
dc.contributor.committeememberKennedy, Thomas Gen_US
dc.contributor.committeememberBhattacharya, Rabindraen_US
dc.contributor.committeememberWatkins, Joseph Cen_US
dc.identifier.proquest11536-
dc.identifier.oclc752261400-
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