Effective Dynamics And Applications Of Stochastic Differential Equations Driven By Tempered Asymmetric Lévy Process

The real world is accompanied by random fluctuations universally.With the perturbance of noise,the evolution and development of things can be accurately described by the stochastic differential equations.Brownian motion is commonly used to simulate random factors.However,a large number of facts have shown that there are sudden,intermittent and unpredictable behaviors in complex systems,such as gene expression regulation.Lévy process is one of the most effective ways to describe these non-Gaussian phenomena.Meanwhile,stationary processes are often required to have finite variance in physical space,while Lévy process do not have second moment.To do that,tempering effects are considered.Mean First Exit Time(MFET)and First Escape Probability(FEP)are effective tools for describing the transitions in stochastic multistable systems.In this thesis,we mainly study the transitions in stochastic differential equations driven by Brownian motion and tempered asymmetric -stable Lévy process.The details are as follows:In chapter 1,we introduce the background of transitions,the definition and properties of Brownian motion and Lévy process,and the domestic and foreign developments.In chapter 2,we introduce the definition of MFET and FEP,point out they satisfy some nonlocal elliptic equation and explain the difficulty to obtain its analytical solution.The highdimensional numerical algorithm and the difference equations are given.In particular,the coefficient matrix in one-dimensional case is decomposed to tridiagonal matrices and a Toeplitz matrix.In chapter 3,we apply the numerical algorithm to a one-dimensional bistable gene regulation network,analyze the corresponding MFET and FEP qualitatively,and study the influence of noise intensities,stable index,skew parameter and tempering index.The results show that both Brownian motion and Lévy process can induce the transitions between adjacent regions in a finite time with a specific probability.Brownian motion cannot induce the transitions between non-adjacent regions,which can be induced by a Lévy process.Asymmetric Lévy process is beneficial to the transitions when the direction is the same as the skew of Lévy processIn chapter 4,a three-dimensional bistable chaotic system with a chaotic attractor and a stable equilibrium is considered.Because of the irregular topology of the chaotic attractor,we analyze the transitions between chaotic attractor and stable equilibrium by the MFET and FEP of the points nearest and farthest from the stable equilibrium.The results show that the transitions from chaotic attractor to a stable equilibrium depend on the initial position.With the same parameters,the nearest point has a shorter MFET and a larger FEP than the farthest point.Finally,we summary this dissertation,point out the shortcomings and present the future topics.