Noise-Induced Transition Phenomena Of Stochastic Dynamical Systems

This thesis studies the transition phenomena of stochastic dynamical systems with Gaussian Brownian motion or non-Gaussian Levy motion.Under the influence of the random factors,the systems exhibit certain different behaviors compared with the deterministic systems.One of the interesting and important phenomena is the transitions between two metastable states.There are three problems discussed in this thesis:1,the equivalent descriptions and characterizations of the most probable transition paths in stochastic dynamical systems with Brownian motion;2,the estimate problem of the most probable transition time of stochastic dynamical systems with Brownian motion;3,the geometrical characterizations of the most probable transition paths in stochastic dynamical systems with Levy motion.This thesis is organized as follows.Chapter 1 introduces the research background,current research status and main research contents.Chapter 2 reviews the concepts about Brownian motion,Levy motion and introduces some basic results about stochastic analysis and stochastic differential equations.Chapter 3 considers a class of stochastic dynamical systems with Brownian motion.The Markovian bridge processes and the SDE representations of Markovian bridges are introduced.The relationship between the bridge measures and Onsager-Machlup action functionals is shown by developing a new method.Based on this relationship,the equivalent descriptions of the most probable transition paths in different forms are proved.These results indicate that,the most probable transition paths can be determined by first order differential equations under certain assumptions.For general nonlinear stochastic dynamics with small noise,the most probable transition paths can be approximated by first order differential equations or integro differential equations.These descriptions and characterizations of the most probable transition paths provide us new results and insights on related topics over the existing methods.After that,the most probable transition time of stochastic dynamical systems with Browinian motion is considered.Instead of minimizing the Onsager-Machlup action functional,the maximum probability that the solution process of the system stays in a neighborhood(or a tube)of a transition path is examined.The lower bound for an exponential decay and the upper bound for a power law decay for the maximum of tube set probabilities are established.Based on these estimates,the lower and upper bounds for the most probable transition time are derived.Then a method to estimate the most probable transition time is devised,and additionally the feasibility and rationality of this estimating method are shown by two examples.Chapter 4 discusses a class of stochastic dynamical systems with non-Gaussian Levy motion.Because the theory of Onsager-Machlup method for non-Gaussian systems is incomplete,the Path Integral method is applied because of its efficiency and simplicity.The geometrical characterization of most probable transition paths is that they have and only have one big jump.This result is shown by investigating the effect of the concavity of theα-stable Lévy motions’ distribution functions.Chapter 5 summarizes the research results,and outlines the follow-up research topics and directions.