無界域上一類隨機反應擴散方程不變測度的存在唯一性

鄧海斌　李曉軍

摘要：研究定義在無界區域上的一類隨機反應擴散方程不變測度的存在性和唯一性.利用方程主部算子在權空間L2ρ（Rd+）上生成算子半群的指數衰減性，對方程的解進行整體期望有界估計，并得到隨機穩態解的存在性和指數穩定性，進而得到穩態解的分布為唯一的不變測度.

關鍵詞：隨機反應擴散方程; 不變測度; 指數穩定

中圖分類號：O175.26 文獻標志碼：A 文章編號：1001-8395（2023）05-0608-08

1相關引理和概念

2穩態解的指數穩定性和一致有界性

3不變測度的存在唯一性

參考文獻

[1] DA PRATO G， ZABCZYK J. Ergodicity for Infinite Dimensional Systems[M]. Cambridge：Cambridge University Press，1996.

[2] DA PRATO G， ZABCZYK J. Stochastic Equations in Infinite Dimensions[M]. Cambridge：Cambridge University Press，1992.

[3] CERRAI S. Stochastic reaction-diffusion systems with multiplicative noise and non-Lipschitz reaction term[J]. Probability Theory and Related Fields，2003，125（2）：271-304.

[4] WANG B X. Dynamics of fractional stochastic reaction-diffusion equations on unbounded domains driven by nonlinear noise[J].Journal of Differential Equations，2019，268（1）：1-59.

[5] MISIATS O， STANZHYTSKYI O， YIP N K. Existence and uniqueness of invariant measures for stochastic reaction-diffusion equations in unbounded domains[J]. Journal of Theoretical Probability，2016，29（3）：996-1026.

[6] ASSING S， MANTHEY R. Invariant measures for stochastic heat equations with unbounded coefficients[J]. Stochastic Processes and Their Applications，2003，103（2）：237-256.

[7] BRZEZNIAK Z， MOTYL E， ONDREJAT M. Invariant measure for the stochastic Navier-Stokes equations in unbounded 2D domains[J]. The Annals of Probability，2017，45（5）：3145-3201.

[8] ECKMANN J， HAIRER M. Invariant measures for stochastic partial differential equations in unbounded domains[J]. Nonlinearity，2001，14（1）：133-151.

[9] TESSITORE G， ZABCZYK J. Invariant measures for stochastic heat equations[J]. Probability and Mathematical Statistics，1998，18（2）：271-287.

[10] BRZENIAK Z， ONDREJT M， SEIDLER J. Invariant measures for stochastic nonlinear beam and wave equations[J]. Journal of Differential Equations，2016，260（5）：4157-4179.

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）