有界区间上随机分数阶反应扩散方程鞅解的存在性

The Existence of Martingale Solution of the Stochastic Fractional Reaction-diffusion Equation on Bounded Intervals

有界区间上随机分数阶反应扩散方程在分数阶非相对量子力学中起到很重要的作用.由于噪声和有界区间上分数阶Laplace算子的扰动和影响,使随机分数阶反应扩散方程的研究变得复杂.通过引入一个适当的加权函数来构造加权空间,运用算子理论来克服有界区间上的分数阶Laplace算子带来的困难.运用Prokhorov定理和Skorokhod嵌入定理来解决噪声带给系统的通常紧性不成立的收敛问题.利用It公式和一系列精致的不等式技巧,以及Galerkin方法,最终获得系统鞅解的存在性.

This paper studies the stochastic fractional reaction-diffusion equation on bounded intervals,which plays a very important role in fractional nonrelativistic quantum mechanics.Due to interacting and disturbing of the fractional Laplacian operator on a bounded interval with white noise,the stochastic fractional reaction-diffusion equation is too complicated to be understood.By introducing a suitable weight function to construct the weighted space,we apply the operator theory to overcome the difficulties caused by the fractional Laplacian operator on bounded intervals.We apply Prokhorov theorem and Skorokhod embedding theorem to solve the convergence problem instead of losing the common compactness of the system caused by the white noise.On the basis of It？ formula,a series of exquisite inequalities and Galerkin approximation,we finally establish the existence of the martingale solution of the stochastic fractional reaction-diffusion equation on bounded intervals.