無(wú)界域上一類隨機(jī)反應(yīng)擴(kuò)散方程不變測(cè)度的存在唯一性

鄧海斌　李曉軍

摘要：研究定義在無(wú)界區(qū)域上的一類隨機(jī)反應(yīng)擴(kuò)散方程不變測(cè)度的存在性和唯一性.利用方程主部算子在權(quán)空間L2ρ（Rd+）上生成算子半群的指數(shù)衰減性，對(duì)方程的解進(jìn)行整體期望有界估計(jì)，并得到隨機(jī)穩(wěn)態(tài)解的存在性和指數(shù)穩(wěn)定性，進(jìn)而得到穩(wěn)態(tài)解的分布為唯一的不變測(cè)度.

關(guān)鍵詞：隨機(jī)反應(yīng)擴(kuò)散方程; 不變測(cè)度; 指數(shù)穩(wěn)定

中圖分類號(hào)：O175.26 文獻(xiàn)標(biāo)志碼：A 文章編號(hào)：1001-8395（2023）05-0608-08

1相關(guān)引理和概念

2穩(wěn)態(tài)解的指數(shù)穩(wěn)定性和一致有界性

3不變測(cè)度的存在唯一性

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Existence and Uniqueness of Invariant Measures for a Class of

Stochastic Reaction-Diffusion Equations on Unbounded DomainsDENG Haibin，LI Xiaojun（College of Science， Hohai University， Nanjing 211100， Jiangsu）

Abstract：In this paper， we study the existence and uniqueness of invariant measures for a class of stochastic reaction-diffusion equations defined on unbounded domains. Using the exponential decay of the operator semigroup generated by the linear operator of the equation on the weight space L2ρ（Rd+）， we get the global boundness of expectation estimation of solution， obtain the existence and exponential stability of the stochastic stationary solution， and deduce that the distribution of the stationary solution is the unique invariant measure.

Keywords：stochastic reaction-diffusion equations; invariant measure; exponential stability 〖=〗

2020 MSC：35K57; 60H15

（編輯 余毅）