随机微分方程解的稳定性和矩有界性

项目名称： 随机微分方程解的稳定性和矩有界性

项目编号： No.11501158

项目类型： 青年科学基金项目

立项/批准年度： 2016

项目学科： 数理科学和化学

项目作者： 王珍

作者单位： 合肥工业大学

项目金额： 18万元

中文摘要： 本项目将研究线性随机时滞微分方程和随机年龄结构模型解的稳定性和矩有界性。首先，对二维具有离散时滞和分布时滞的线性随机微分方程，深入细致地分析解的二阶矩对应的特征方程根的分布情况，建立便于应用的二阶矩有界和无界的充分条件。作为理论结果的应用，本项目还将研究具有离散时滞和分布时滞随机捕食者-食饵模型的稳定性和矩有界性。其次，对线性随机年龄结构模型和随机时滞年龄结构模型，将利用Laplace变换、Lyapunov泛函以及一些经典不等式，研究其解的随机稳定性和矩有界性，以及出生系数、死亡系数和时滞等参数和随机扰动对解的随机稳定性和矩有界性的影响。

中文关键词： 随机稳定性；矩有界性；特征方程；随机时滞微分方程；随机年龄结构模型

英文摘要： In this project, we will study the stochastic stability and the boundedness of moments of the solutions to linear stochastic delay differential equations and stochastic age-structured models. First, for the 2-dimesional linear stochastic differential equations with a discrete delay and the linear stochastic differential equations with the distributed delay, we will analyze the distribution of the roots of the characteristic equations for the second moments of their solutions, and will give the sufficient conditions for boundedness and unboundedness of the moments, respectively. As the application of theoretic results, we will investigate the stochastic stability and the moment boundedness for the stochastic predator-prey model with a discrete delay and the distributed delay, respectively. Second, for stochastic linear age-structured model and stochastic delay age-structured model, we will study the stochastic stability and the boundedness of moments of their solutions through the technique of the Laplace transforms, Lyapunov functionals and some classical inequalities. Moreover, we will also investigate the influence of some parameters, such as morality modulus, birth modulus and time delay, and stochastic perturbations on the stochastic stability and the moment boundedness.

英文关键词： stochastic stability;moment boundedness;characteristic equation;stochastic delay differential equation; stochastic age-structured model