Large Deviations For Stochastic Differential Equations With Jumps And Applications To Transition Paths

The noise term in stochastic differential equations will cause transitions between sta-ble states,which will not happen for deterministic differential equations.For this reason,stochastic differential equations are widely used in the investigation of transition problems.This dissertation investigates the large deviations theory that can be used to quantify the tran-sition behaviors in stochastic systems.The dissertation also applies and develops numerical methods to compute the most likely transition paths for stochastic systems.The dissertation deals with stochastic differential equations driven by Lévy processes and semimartingales.Firstly,the dissertation proves the large deviations results for a class of one-dimensional stochastic differential equations driven by heavy-tailed Lévy processes.In the existing stud-ies,due to the relationship between the uniformly exponential tightness and the large devia-tions principle,the driving Lévy processes are inevitably required to be exponentially inte-grable and the heavy-tailed Lévy processes do not satisfy such condition.Therefore,based on the results of the heavy-tailed Lévy process and the idea of the contraction principle,the large deviations for the stochastic differential equations driven by such heavy-tailed Lévy processes are proved through a proper solution mapping.It is also proved that such stochas-tic differential equations only satisfy the weak large deviations principle with a logarithmic speed.As an application of this result,the large deviations for the stochastic differential equations driven by α-stable Lévy processes are also proved for the first time.Secondly,the dissertation proposes a stochastic differential equation on a Riemannian manifold to model the dynamics of a Riemannian manifold rolling along a given trajectory.And the exponential stability of this dynamical model is examined through the large devi-ations principle.The difference from the current studies is that the dissertation keeps the random twisting in this model and includes random slipping through semimartingales.For both compact and non-compact manifolds,the large deviations principle is proved for the stochastic differential equation driven by semimartingales on the orthogonal frame bundle,based on the ralationship between the uniformly exponential tightness and the large devia-tions principle.Moreover,its projection on the base manifold and the horizontal lift of this projection are proved to satisfy the large deviations principle.In the Euclidean space,the large deviations principle for a class of stochastic differential equations driven by continuous semimartingales is also proved.Thirdly,as an application of the large deviations principle,the dissertation applies the geometric minimum action method to calculate the most likely transition paths from the steady state to the oscillatory state in a stochastic oceanic carbon cycle model for different additional carbon dioxide input intensities.Based on these results,an early warning sign for the dramatic change in the carbonate state of the ocean is obtained.Finally,the dissertation proposes a new numerical algorithm to compute the most likely transition paths for stochastic systems under non-Gaussian Lévy noise.The related action functionals for stochastic systems under non-Gaussian Lévy noise can only be formulated as the minimizer of a constrained minimization problem,which can not be simplified to the integral of an explicit function as in the Gaussian case.This makes the existing numerical methods not applicable here.As a result of that,the dissertation proposes an optimal control problem and obtains the most likely transition path through training neural networks with properly constructed loss functions.Moreover,the reliability and validity of the method are shown by comparing it with the known results under Gaussian noise.