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Title: Flows of stochastic dynamical systems : ergodic theory of stochastic flows
Author: Carverhill, Andrew
ISNI:       0000 0001 3524 5348
Awarding Body: University of Warwick
Current Institution: University of Warwick
Date of Award: 1983
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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.
Supervisor: Not available Sponsor: Science and Engineering Research Council
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: QA Mathematics