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Flows of stochastic dynamical systems : ergodic theory of stochastic flows
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Carverhill, Andrew (1983) Flows of stochastic dynamical systems : ergodic theory of stochastic flows. PhD thesis, University of Warwick.

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Official URL: http://webcat.warwick.ac.uk/record=b3253837~S15
Abstract
In this thesis we present results and examples concerning the asymptotic (large time) behaviour of the flow of a nondegenerate smooth stochastic dynamical system on a smooth compact manifold.
In Chapter 2 we prove a stochastic version of the Oseledec (Multiplicative Ergodic) Theorem for flows (theorem 2.1), in which we define the Lyapunov spectrum for the stochastic flow. Then we obtain stochastic analogies (Theorems 2.2.1, 2.2.2) of the Stable Manifold Theorems of Ruelle [16]. These theorems are proved by adapting Ruelle'S techniques to our situation. Also we discuss the implications of ‘Lyapunov stability', which we define to be the situation when the Lyapunov spectrum is strictly negative. In this situation the trajectories of the flow cluster in a certain way. (Proposition 2.3.3)
In Chapter 3 we give some examples of systems for which we can calculate the Lyapunov spectrum. We can choose our parameters such that these systems are Lyapunov stable, and in this case we can calculate the flows and their asymptotic behaviour completely.
In Chapter 4 we give a formula for the Lyapunov numbers which is analogous to that of Khas’ninskii [9] for a linear system. Then we use this formula to prove a theorem on the preservation of Lyapunov stability under a stochastic perturbation.
Item Type:  Thesis (PhD)  

Subjects:  Q Science > QA Mathematics  
Library of Congress Subject Headings (LCSH):  Stochastic systems, Differentiable dynamical systems, Ergodic theory, Manifolds (Mathematics), Lyapunov stability  
Official Date:  February 1983  
Dates: 


Institution:  University of Warwick  
Theses Department:  Mathematics Institute  
Thesis Type:  PhD  
Publication Status:  Unpublished  
Supervisor(s)/Advisor:  Elworthy, K. D.  
Sponsors:  Science and Engineering Research Council (Great Britain)  
Format of File:  
Extent:  89 leaves  
Language:  eng 
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