一類半線性隨機微分方程的均方漸近概自守溫和解

姚慧麗　霍貴珍　孫海彤　王晶囡

摘要：均方概自守型函數理論在隨機微分方程中的應用越來越引起數學研究者的關注，這類方程的均方漸近概自守解比均方概自守解的應用范圍更加廣泛。對一類半線性隨機微分方程的均方漸近概自守溫和解進行探討。利用Banach壓縮映射原理，結合均方漸近概自守隨機過程的定義和性質、Cauchy-Schwarz不等式、Lipschitz條件、It等距積分，討論了該類隨機微分方程的均方漸近概自守溫和解的存在唯一性。

關鍵詞：均方漸近概自守溫和解;半線性隨機微分方程;Banach壓縮映射原理

DOI：10.15938/j.jhust.2022.04.020

中圖分類號： O175

文獻標志碼： A

文章編號： 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

概自守函數、漸近概自守函數以及偽概自守函數（統稱為概自守型函數）的定義分別由BOCHNER S、N′GUEREKATA G M、XIAO T J， LIANG J， ZHANG J給出[1-3]。概自守型函數理論的產生推廣了概周期型函數的應用范圍，并在各類方程中得到了應用[4-10]，為了更好的描述自然界中的隨機現象，2010年，FU M M， LIU Z X提出了均方概自守隨機過程的概念[11]，這一概念是對概自守函數的推廣。之后，均方偽概守隨機過程和均方漸近概自守隨機過程的概念也相繼被給出[ 12-13 ] 。自均方概自守型隨機過程有關理論被提出以來，國內外數學工作者將其應用到一類將隨機性納入了數學描述中的模型中即隨機微分方程中，研究了此種方程的均方概自守解[14-16]和均方偽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.對下列一類半線性隨機微分方程

1預備知識

2主要結論

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（編輯：溫澤宇）