Some Ergodic Theorems for a Parabolic Anderson Model

作     者：Yong LIU LMAM Feng Xia YANG 

作者机构：School of Mathematical Sciencesand Institute of Mathematicsand Center for Statistical SciencePeking University LMAMSchool of Mathematical SciencesPeking University 

出 版 物：《Acta Mathematica Sinica,English Series》 (数学学报（英文版）)

年 卷 期：2012年第28卷第12期

页      面：2443-2462页

核心收录：

学科分类：02[经济学] 0202[经济学-应用经济学] 020208[经济学-统计学] 07[理学] 0714[理学-统计学（可授理学、经济学学位）] 070103[理学-概率论与数理统计] 0701[理学-数学] 

基　　金：The first author is supported by National Natural Science Foundation of China (Grant Nos. 10531070,11071008) SRF for ROCS,Science and Technology Ministry 973 project (2006CB805900) the Doctoral Program Foundation of the Ministry of Education,China 

主　　题：Linear system of interacting diffusion parabolic Anderson model ergodic invariant measures clustering phenomena 

摘      要：In this paper, we study some ergodic theorems of a class of linear systems of interacting diffusions, which is a parabolic Anderson model. First, under the assumption that the transition kernel a = （a（i,j））i,j∈s is doubly stochastic, we obtain the long-time convergence to an invariant probability measure Vh starting from a bounded a-harmonic function h based on self-duality property, and then we show the convergence to the invariant probability measure Uh holds for a broad class of initial distributions. Second, if （a（i, j））i,j∈s is transient and symmetric, and the diffusion parameter c remains below a threshold, we are able to determine the set of extremal invariant probability measures with finite second moment. Finally, in the case that the transition kernel （a（i,j））i,j∈s is doubly stochastic and satisfies Case I （see Case I in [Shiga, T.： An interacting system in population genetics. J. Math. Kyoto Univ., 20, 213-242 （1980）]）, we show that this parabolic Anderson model locally dies out independent of the diffusion parameter c.