姚慧麗　霍貴珍　孫海彤　王晶囡

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

關(guān)鍵詞：均方漸近概自守溫和解;半線性隨機(jī)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(xiàn)標(biāo)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(shù)、漸近概自守函數(shù)以及偽概自守函數(shù)（統(tǒng)稱為概自守型函數(shù)）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(shù)理論的產(chǎn)生推廣了概周期型函數(shù)的應(yīng)用范圍，并在各類方程中得到了應(yīng)用[4-10]，為了更好的描述自然界中的隨機(jī)現(xiàn)象，2010年，F(xiàn)U M M， LIU Z X提出了均方概自守隨機(jī)過程的概念[11]，這一概念是對概自守函數(shù)的推廣。之后，均方偽概守隨機(jī)過程和均方漸近概自守隨機(jī)過程的概念也相繼被給出[ 12-13 ] 。自均方概自守型隨機(jī)過程有關(guān)理論被提出以來，國內(nèi)外數(shù)學(xué)工作者將其應(yīng)用到一類將隨機(jī)性納入了數(shù)學(xué)描述中的模型中即隨機(jī)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(jī)微分方程

1預(yù)備知識

2主要結(jié)論

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（編輯：溫澤宇）