On Existence And Stability Of Solutions To Stochastic Differential Systems

This thesis is devoted to the study of stochastic differential systems in two aspects:coupled stochastic nonlinear Schr(?)dinger problem in variational formulation and existence and stability of mild solutions to random impulsive stochastic differential system with non-compact semigroups.The first part is to discuss a class of coupled stochastic nonlinear Schr(?)dinger system with linear multiplicative Gaussian noise and power-nonlinear terms,which is also called coupled stochastic Gross–Pitaevskii systems.The main motivation comes from the physi-cal significance of the energy space H~1,such as the Bose–Einstein condensates,nonlinear optics field and so on.We consider the focusing nonlinearities and attractive coupled nonlin-ear terms in the stochastic system,and we prove the existence and uniqueness of variational solutions to the stochastic problem in the inner product space L~2and energy space H~1,respectively.Our approach is different from the previous literature on coupled stochastic nonlinear Schr(?)dinger systems.We firstly reduce such a stochastic system to a coupled de-terministic nonlinear Schr(?)dinger system,which greatly simplifies the stochastic problem.Secondly,we treat the deterministic system by Galerkin approximation argument,where we combine energy identities with convergence theory to show the existence and prior estima-tions of variational solutions to the finite-dimensional Galerkin system.Thirdly,according to the It(?)’s product formula together with the previous results,we prove the existence of variational solutions of the coupled stochastic nonlinear Schr(?)dinger system.The second part is to discuss a class of stochastic differential systems with random impulses which arises from the real-world significance,such as biological models involving thresholds,stock pricing models in finance,optimal control models in economy and so on.We consider the semigroups that are noncompact and thus we combine the noncompact Hausdorff measure theory and M(?)nch fixed point theorem to prove the existence of the mild solution of the systems.Furthermore,we explore two kinds of stabilities of mild solutions for the system by employing the Lipschitz condition and Gronwall’s inequality,where one of them describes the continuous dependence to the initial data,and the other is the Hyers–Ulam stability.Finally,we prove the mean-square exponential stability of the mild solutions through an impulsive integral inequality,and give an abstract example as a reference.