一类半线性随机微分方程的均方渐近概自守温和解

姚慧丽　霍贵珍　孙海彤　王晶囡

摘要：均方概自守型函數理论在随机微分方程中的应用越来越引起数学研究者的关注，这类方程的均方渐近概自守解比均方概自守解的应用范围更加广泛。对一类半线性随机微分方程的均方渐近概自守温和解进行探讨。利用Banach压缩映射原理，结合均方渐近概自守随机过程的定义和性质、Cauchy-Schwarz不等式、Lipschitz条件、It等距积分，讨论了该类随机微分方程的均方渐近概自守温和解的存在唯一性。

关键词：均方渐近概自守温和解;半线性随机微分方程;Banach压缩映射原理

DOI：10.15938/j.jhust.2022.04.020

中图分类号： O175

文献标志码： A

文章编号： 1007-2683（2022）04-0154-07

Square-Mean Asymptotically Almost Automorphic Mild Solutions

to a Class of Semi-linear Stochastic Differential Equations

YAO Hui-li，HUO Gui-zhen，SUN Hai-tong，WANG Jing-nan

（School of Science，Harbin University of Science and Technology，Harbin 150080，China）

Abstract：The applications of the theories of square-mean almost automorphic type functions have attracted more and more attention by mathematics researchers， square-mean asymptotically almost automorphic solutions of this class of differential equations have a wider range of applications than square-mean almost automorphic solutions.Square-mean asymptotically almost automorphic mild solutions to a class of semi-linear stochastic differential equations are investigated. The existence and uniqueness of square-mean asymptotically almost automorphic mild solutions for this kind of equation are discussed by using the principle of Banach compressed image， combining with the definition and properties of square-mean asymptotically almost automorphic stochastic processes， Cauchy-Schwarz inequality， Lipschtiz conditions and Ito integrals isometry.

Keywords：square-mean asymptotically almost automorphic mild solutions; semi-linearstochastic differential equations; principle of Banach compressed image

0引言

概自守函数、渐近概自守函数以及伪概自守函数（统称为概自守型函数）的定义分别由BOCHNER S、N′GUEREKATA G M、XIAO T J， LIANG J， ZHANG J给出[1-3]。概自守型函数理论的产生推广了概周期型函数的应用范围，并在各类方程中得到了应用[4-10]，为了更好的描述自然界中的随机现象，2010年，FU M M， LIU Z X提出了均方概自守随机过程的概念[11]，这一概念是对概自守函数的推广。之后，均方伪概守随机过程和均方渐近概自守随机过程的概念也相继被给出[ 12-13 ] 。自均方概自守型随机过程有关理论被提出以来，国内外数学工作者将其应用到一类将随机性纳入了数学描述中的模型中即随机微分方程中，研究了此种方程的均方概自守解[14-16]和均方伪概自守解的存在及唯一性[17-18]。在文[14]中，CHANG Y K， ZHAO Z H， N′GUEREKATA G M.对下列一类半线性随机微分方程

1预备知识

2主要结论

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（编辑：温泽宇）