Researches On Several Types Of Function Spaces With Stochastic Exponents

Variable exponent function spaces have a good performance in solving some non-linear problems of natural science and engineering. For example, variable exponent Lebesgue and Sobolev spaces play an important role in the study of a class of nonlinear problems with variable exponential growth. With the emergence of nonlinear problems in financial engineering, on which stochastic analysis is applied, such as nonlinear pricing operators and nonlinear expectations, we hope that variable exponent function spaces also have a good performance in some nonlinear problems of financial engineering. Therefore, the study on stochastic variable exponent function spaces and the extension to stochastic analysis are very necessary.In this dissertation, we extend classical variable exponent spaces to stochastic vari-able exponent spaces, to stochastic process spaces with random variable exponents and to variable exponent Lebesgue and Sobolev spaces in the Hilbert space under Gaussian measure. And in these framework, we discuss the properties of spaces and give some applications.The main contents are as follows:1. Stochastic field exponent spaces are introduced. The spaces with a stochastic field exponent Lp(x,ω)(D×Ω) and WKp(x,ω)(D×Ω) are introduced, and then the proper-ties of these spaces, for example, completeness, reflexivity, embedding theorem etc, are discussed. Finally, we get existence and uniqueness of the weak solution for stochastic partial differential equations with stochastic field exponent growth in W01,p(x,ω)(D×Ω).2. Variable exponents are introduced to stochastic process spaces. Spaces with stochastic variable exponents CB([0, T];Lp(·)(Ω)) and LBp(·)(Ω;C([O, T])) are introduced. After discussing the properties of spaces, such as completeness, containment relationship among spaces etc, the moduli of Ito integrals for real stochastic processes in Lp(ω)([0, T]×Ω^,(?)μ×P) are estimated, and we get that Ito integrals in Lp(ω)([O, T]×Ω,(?),μ×P) belong to CB([0, T]; Lp(·)(Ω)). Finally, as an application, we prove the existence and uniqueness of the solution for semilinear stochastic partial differential equations by contraction mapping principle.3. Variable exponent spaces on the Hilbert space H with a Gaussian measure μ are introduced. Some approximation results of Lp(x)(H,μ) are discussed, and we prove that C0(H) is dense in Lp(x)(H,μ) and (?)(H) is dense in Lp(x)(H,μ). By these approxi-mation results and closable operator results, we give extension of Malliavin derivatives from D1,2(H,μ) to the variable exponent Sobolev space D1,p(x)(H,μ). Some properties of Malliavin derivatives in D1,p(x)(H,μ) are also discussed.