Stochastic Dynamic Equations On Time Scales

In this paper, some properties of eigenvalues and solutions of stochastic dynamicequations on time scales will be considered.The paper is mainly divided into four chapters.The first chapter is the introduction of the whole paper. We talk about the back-ground of stochastic dynamic equations.In the second chapter of the paper, we introduce the basic concepts and theories.It can be divided into two sections, at first, the theories of diferential and integralcalculus are introduced, then we describe the basic concepts on the probability space.The third chapter is the main part of the paper, which researches the diferentialequation with random coefcients on time scales. It can be divided into three parts.In part one we obtain random solutions of initial value problems continuously dependon the coefcients in the equation by using inequality of Gro¨nwall. On this basis, weprove the solutions of initial value problems are diferentiable on the coefcients inthe equation. In the above two parts the initial value problems of stochastic dynamicequation on time scales have been shown, then, we consider the boundary value problemsof stochastic dynamic equation in part three. Using the theory of integral equations,we study some properties of eigenvalues and eigenfunctions, and a simple estimate ofrandom eigenvalues will be given.In the fourth part, we mainly investigate a class of relatively simple Sturm-Liouvilleproblem with random boundary conditions, talking about the properties of continuityand diferentiability of the random eigenvalues of Sturm-Liouville problem on the ran-dom variables in the boundary conditions. And we also give the corresponding conver-gence of random eigenvalues when the random variable in the boundary conditions hasa certain convergence.At last, we summarize the results of the whole paper.