Fixed points of smoothing transformation in random environment  

作　　者：Xiaoyue ZHANG Wenming HONG 

机构地区：[1]School of Statistics,Capital University of Economics and Business,Beijing 100070,China [2]School of Mathematical Sciences,Laboratory of Mathematics and Complex Systems,Beijing Normal University,Beijing 100875,China

出　　处：《Frontiers of Mathematics in China》2021年第4期1191-1210,共20页中国高等学校学术文摘·数学（英文）

基　　金：the National Key Research and Development Program of China(No.2020YFA0712900);the National Natural Science Foundation of China(Grant No.11971062);the Scientific Research Foundation for Young Teachers in Capital University of Economics and Business(NO.XRZ2021035).

摘　　要：At each time n∈N,let Y^(n)(ξ)=(y^(n)_(1)(ξ),y^(n)_(2)(ξ),…)be a random sequence of non-negative numbers that are ultimately zero in a random environmentξ=(ξ_(n))n∈N.The existence and uniqueness of the non-negative fixed points of the associated smoothing transformation in random environment are considered.These fixed points are solutions to the distributional equation for a.e.ξ,Z(ξ)=dΣ_(i∈N_(+))y^(0)_(i)(ξ)Z^(1)_(i)(ξ),where{Z^(1)_(i):i∈N_(+)}are random variables in random environment which satisfy that for any environmentξ,under P_(ξ),{Z^(1)_(i)(ξ):i∈N_(+)}are independent of each other and Y^(0)(ξ),and have the same conditional distribution P_(ξ)(Z^(1)_(i)(ξ)∈·)=P_(Tξ)(Z(Tξ)∈·),where T is the shift operator.This extends the classical results of J.D.Biggins[J.Appl.Probab.,1977,14:25-37]to the random environment case.As an application,the martingale convergence of the branching random walk in random environment is given as well.

关 键 词：Smoothing transformation functional equation branching random walk random environment MARTINGALES 

分 类 号：O17[理学—数学]