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dc.contributor.advisor Mattingly, Jonathan C en_US
dc.contributor.author Kolba, Tiffany Nicole en_US
dc.date.accessioned 2012-05-25T20:09:54Z
dc.date.available 2012-05-25T20:09:54Z
dc.date.issued 2012 en_US
dc.identifier.uri http://hdl.handle.net/10161/5444
dc.description Dissertation en_US
dc.description.abstract <p>Noise-induced stabilization occurs when an unstable deterministic system is stabilized by the addition of white noise. Proving that this phenomenon occurs for a particular system is often manifested through the construction of a global Lyapunov function. However, the procedure for constructing a Lyapunov function is often quite ad hoc, involving much time and tedium. In this thesis, a systematic algorithm for the construction of a global Lyapunov function for planar systems is presented. The general methodology is to construct a sequence of local Lyapunov functions in different regions of the plane, where the regions are delineated by different behaviors of the deterministic dynamics. A priming region, where the deterministic drift is directed inward, is first identified where there is an obvious choice for a local Lyapunov function. This priming Lyapunov function is then propagated to the other regions through a series of Poisson equations. The local Lyapunov functions are lastly patched together to form one smooth global Lyapunov function.</p><p>The algorithm is applied to a model problem which displays finite time blow up in the deterministic setting in order to prove that the system exhibits noise-induced stabilization. Moreover, the Lyapunov function constructed is in fact what we define to be a super Lyapunov function. We prove that the existence of a super Lyapunov function, along with a minorization condition, implies that the corresponding system converges to a unique invariant probability measure at an exponential rate that is independent of the initial condition.</p> en_US
dc.subject Mathematics en_US
dc.subject Lyapunov Function en_US
dc.subject Probability en_US
dc.subject Stabilization en_US
dc.subject Stochastic Differential Equation en_US
dc.title A Generalized Lyapunov Construction for Proving Stabilization by Noise en_US
dc.type Dissertation en_US
dc.department Mathematics en_US

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