Numerical Algorithms For Mean Exit Time And Escape Probability Of Non-Gaussian Dynamical Systems

The investigation of stochastic dynamical systems has attracted a lot of attention. During the past several years, there are not only many important breakthroughs in the dynamical systems driven by Brownian motion but also many applications in other fields such as geophysics and biology. However, noisy fluctuations are usually non-Gaussian in nature, and the investigation of non-Gaussian dynamical systems is still in its infancy. The α-stable Levy noise is a special but important class of non-Gaussian noises. In this thesis, we mainly consider the stochastic dynamical systems driven by symmetric and asymmetric α-stable Levy noise.Mean exit time and escape probability are described by nonlocal partial differential equations.First, we study the mean exit time and escape probability for systems driven by asymmetric a-stable type Levy motions, which attracts more and more attentions. We simplify the problem by non-dimensionalization, and then devise a numerical method for computing mean exit time and escape probability by discretizing nonlocal partial differential equations. At last, we examine the effects of the different factors on mean exit time, including the skewness parameter, the size of the domain, the drift term, and the intensity of Gaussian and non-Gaussian noises.Then, we develop a new method to solve the Fokker-Planck equtaion for systems driven by asymmetric α-stable Levy motion, and discuss the corresponding dynamic behavior.Finally, we consider mean exit time and escape probability for systems with two dimensional rotationally symmetric a-stable type Levy motions. After the assumption of radial symmetry for the process and the domains, we exploit the Gauss-Hypergeometric function in designing our numerical method, and validate the convergence of our new methods. The effects of drift, Gaussian noises, intensity of jump measure and domain sizes on the mean exit time are discussed. The difference between the one-dimensional and two-dimensional cases is also presented.